id	sid	tid	token	lemma	pos
ma-287	1	1	2025	2025	NUM
ma-287	1	2	ada	ada	PROPN
ma-287	1	3	academica	academica	PROPN
ma-287	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-287	1	5	.	.	PUNCT
ma-287	2	1	j.	j.	PROPN
ma-287	2	2	math	math	PROPN
ma-287	2	3	.	.	PUNCT
ma-287	3	1	anal	anal	ADJ
ma-287	3	2	.	.	PUNCT
ma-287	4	1	5	5	NUM
ma-287	4	2	(	(	PUNCT
ma-287	4	3	2025	2025	NUM
ma-287	4	4	)	)	PUNCT
ma-287	4	5	12doi	12doi	NUM
ma-287	4	6	:	:	PUNCT
ma-287	4	7	10.28924	10.28924	NUM
ma-287	4	8	/	/	SYM
ma-287	4	9	ada	ada	PROPN
ma-287	4	10	/	/	SYM
ma-287	4	11	ma.5.12	ma.5.12	VERB
ma-287	4	12	some	some	DET
ma-287	4	13	new	new	ADJ
ma-287	4	14	series	series	NOUN
ma-287	4	15	expansions	expansion	NOUN
ma-287	4	16	of	of	ADP
ma-287	4	17	a	a	DET
ma-287	4	18	special	special	ADJ
ma-287	4	19	type	type	NOUN
ma-287	4	20	of	of	ADP
ma-287	4	21	functions	function	NOUN
ma-287	4	22	involving	involve	VERB
ma-287	4	23	the	the	DET
ma-287	4	24	logarithmic	logarithmic	ADJ
ma-287	4	25	function	function	NOUN
ma-287	4	26	christophe	christophe	PROPN
ma-287	4	27	chesneau	chesneau	PROPN
ma-287	4	28	department	department	PROPN
ma-287	4	29	of	of	ADP
ma-287	4	30	mathematics	mathematics	PROPN
ma-287	4	31	,	,	PUNCT
ma-287	4	32	lmno	lmno	NOUN
ma-287	4	33	,	,	PUNCT
ma-287	4	34	university	university	NOUN
ma-287	4	35	of	of	ADP
ma-287	4	36	caen	caen	PROPN
ma-287	4	37	-	-	PUNCT
ma-287	4	38	normandie	normandie	PROPN
ma-287	4	39	,	,	PUNCT
ma-287	4	40	14032	14032	NUM
ma-287	4	41	caen	caen	PROPN
ma-287	4	42	,	,	PUNCT
ma-287	4	43	france	france	PROPN
ma-287	4	44	christophe.chesneau@gmail.com	christophe.chesneau@gmail.com	X
ma-287	4	45	abstract	abstract	ADJ
ma-287	4	46	.	.	PUNCT
ma-287	5	1	in	in	ADP
ma-287	5	2	this	this	DET
ma-287	5	3	article	article	NOUN
ma-287	5	4	,	,	PUNCT
ma-287	5	5	we	we	PRON
ma-287	5	6	present	present	VERB
ma-287	5	7	new	new	ADJ
ma-287	5	8	series	series	NOUN
ma-287	5	9	expansions	expansion	NOUN
ma-287	5	10	for	for	ADP
ma-287	5	11	a	a	DET
ma-287	5	12	certain	certain	ADJ
ma-287	5	13	family	family	NOUN
ma-287	5	14	of	of	ADP
ma-287	5	15	functions	function	NOUN
ma-287	5	16	thatdepend	thatdepend	VERB
ma-287	5	17	on	on	ADP
ma-287	5	18	the	the	DET
ma-287	5	19	logarithmic	logarithmic	ADJ
ma-287	5	20	function	function	NOUN
ma-287	5	21	.	.	PUNCT
ma-287	6	1	a	a	DET
ma-287	6	2	general	general	ADJ
ma-287	6	3	result	result	NOUN
ma-287	6	4	is	be	AUX
ma-287	6	5	demonstrated	demonstrate	VERB
ma-287	6	6	by	by	ADP
ma-287	6	7	considering	consider	VERB
ma-287	6	8	a	a	DET
ma-287	6	9	tunable	tunable	ADJ
ma-287	6	10	inter	inter	ADJ
ma-287	6	11	-	-	ADJ
ma-287	6	12	mediate	mediate	ADJ
ma-287	6	13	function	function	NOUN
ma-287	6	14	.	.	PUNCT
ma-287	7	1	this	this	DET
ma-287	7	2	result	result	NOUN
ma-287	7	3	has	have	VERB
ma-287	7	4	the	the	DET
ma-287	7	5	interest	interest	NOUN
ma-287	7	6	of	of	ADP
ma-287	7	7	unifying	unify	VERB
ma-287	7	8	several	several	ADJ
ma-287	7	9	important	important	ADJ
ma-287	7	10	results	result	NOUN
ma-287	7	11	in	in	ADP
ma-287	7	12	the	the	DET
ma-287	7	13	literature	literature	NOUN
ma-287	7	14	,	,	PUNCT
ma-287	7	15	including	include	VERB
ma-287	7	16	a	a	DET
ma-287	7	17	well	well	ADV
ma-287	7	18	-	-	PUNCT
ma-287	7	19	known	know	VERB
ma-287	7	20	series	series	NOUN
ma-287	7	21	expansion	expansion	NOUN
ma-287	7	22	established	establish	VERB
ma-287	7	23	by	by	ADP
ma-287	7	24	srinivasa	srinivasa	PROPN
ma-287	7	25	ramanujan	ramanujan	PROPN
ma-287	7	26	.	.	PUNCT
ma-287	8	1	several	several	ADJ
ma-287	8	2	precise	precise	ADJ
ma-287	8	3	ex	ex	NOUN
ma-287	8	4	-	-	ADJ
ma-287	8	5	amples	ample	NOUN
ma-287	8	6	are	be	AUX
ma-287	8	7	given	give	VERB
ma-287	8	8	and	and	CCONJ
ma-287	8	9	discussed	discuss	VERB
ma-287	8	10	in	in	ADP
ma-287	8	11	detail	detail	NOUN
ma-287	8	12	.	.	PUNCT
ma-287	9	1	in	in	ADP
ma-287	9	2	addition	addition	NOUN
ma-287	9	3	,	,	PUNCT
ma-287	9	4	we	we	PRON
ma-287	9	5	recover	recover	VERB
ma-287	9	6	the	the	DET
ma-287	9	7	so	so	ADV
ma-287	9	8	-	-	PUNCT
ma-287	9	9	called	call	VERB
ma-287	9	10	seidel	seidel	PROPN
ma-287	9	11	formula	formula	NOUN
ma-287	9	12	andderive	andderive	ADJ
ma-287	9	13	new	new	ADJ
ma-287	9	14	product	product	NOUN
ma-287	9	15	expansions	expansion	NOUN
ma-287	9	16	,	,	PUNCT
ma-287	9	17	with	with	ADP
ma-287	9	18	an	an	DET
ma-287	9	19	emphasis	emphasis	NOUN
ma-287	9	20	on	on	ADP
ma-287	9	21	the	the	DET
ma-287	9	22	so	so	ADV
ma-287	9	23	-	-	PUNCT
ma-287	9	24	called	call	VERB
ma-287	9	25	einstein	einstein	PROPN
ma-287	9	26	function	function	NOUN
ma-287	9	27	.	.	PUNCT
ma-287	10	1	some	some	DET
ma-287	10	2	inequali	inequali	ADJ
ma-287	10	3	-	-	PUNCT
ma-287	10	4	ties	tie	NOUN
ma-287	10	5	involving	involve	VERB
ma-287	10	6	logarithmic	logarithmic	ADJ
ma-287	10	7	functions	function	NOUN
ma-287	10	8	are	be	AUX
ma-287	10	9	also	also	ADV
ma-287	10	10	applications	application	NOUN
ma-287	10	11	of	of	ADP
ma-287	10	12	our	our	PRON
ma-287	10	13	series	series	NOUN
ma-287	10	14	expansion	expansion	NOUN
ma-287	10	15	approach	approach	NOUN
ma-287	10	16	.	.	PUNCT
ma-287	11	1	selectedresults	selectedresult	NOUN
ma-287	11	2	are	be	AUX
ma-287	11	3	supported	support	VERB
ma-287	11	4	by	by	ADP
ma-287	11	5	graphical	graphical	ADJ
ma-287	11	6	work	work	NOUN
ma-287	11	7	.	.	PUNCT
ma-287	12	1	1	1	X
ma-287	12	2	.	.	X
ma-287	12	3	introduction	introduction	NOUN
ma-287	12	4	the	the	DET
ma-287	12	5	logarithmic	logarithmic	ADJ
ma-287	12	6	function	function	NOUN
ma-287	12	7	,	,	PUNCT
ma-287	12	8	denoted	denote	VERB
ma-287	12	9	log(x	log(x	PROPN
ma-287	12	10	)	)	PUNCT
ma-287	12	11	,	,	PUNCT
ma-287	12	12	plays	play	VERB
ma-287	12	13	a	a	DET
ma-287	12	14	crucial	crucial	ADJ
ma-287	12	15	role	role	NOUN
ma-287	12	16	in	in	ADP
ma-287	12	17	several	several	ADJ
ma-287	12	18	mathematical	mathematical	ADJ
ma-287	12	19	contexts	contexts	NOUN
ma-287	12	20	,	,	PUNCT
ma-287	12	21	including	include	VERB
ma-287	12	22	calculus	calculus	NOUN
ma-287	12	23	,	,	PUNCT
ma-287	12	24	number	number	NOUN
ma-287	12	25	theory	theory	NOUN
ma-287	12	26	,	,	PUNCT
ma-287	12	27	and	and	CCONJ
ma-287	12	28	computer	computer	NOUN
ma-287	12	29	science	science	NOUN
ma-287	12	30	.	.	PUNCT
ma-287	13	1	see	see	VERB
ma-287	13	2	[	[	X
ma-287	13	3	13	13	NUM
ma-287	13	4	]	]	PUNCT
ma-287	13	5	,	,	PUNCT
ma-287	13	6	and	and	CCONJ
ma-287	13	7	the	the	DET
ma-287	13	8	references	reference	NOUN
ma-287	13	9	therein.understanding	therein.understande	VERB
ma-287	13	10	its	its	PRON
ma-287	13	11	properties	property	NOUN
ma-287	13	12	,	,	PUNCT
ma-287	13	13	especially	especially	ADV
ma-287	13	14	its	its	PRON
ma-287	13	15	various	various	ADJ
ma-287	13	16	series	series	NOUN
ma-287	13	17	expansions	expansion	NOUN
ma-287	13	18	,	,	PUNCT
ma-287	13	19	is	be	AUX
ma-287	13	20	fundamental	fundamental	ADJ
ma-287	13	21	to	to	ADP
ma-287	13	22	manymathematical	manymathematical	ADJ
ma-287	13	23	analyses	analysis	NOUN
ma-287	13	24	.	.	PUNCT
ma-287	14	1	the	the	DET
ma-287	14	2	classical	classical	ADJ
ma-287	14	3	(	(	PUNCT
ma-287	14	4	taylor	taylor	PROPN
ma-287	14	5	)	)	PUNCT
ma-287	14	6	expansion	expansion	NOUN
ma-287	14	7	of	of	ADP
ma-287	14	8	log(x	log(x	PROPN
ma-287	14	9	)	)	PUNCT
ma-287	14	10	is	be	AUX
ma-287	14	11	given	give	VERB
ma-287	14	12	by	by	ADP
ma-287	14	13	log(x	log(x	PROPN
ma-287	14	14	)	)	PUNCT
ma-287	14	15	=	=	PUNCT
ma-287	15	1	+	+	PUNCT
ma-287	15	2	∞∑	∞∑	NUM
ma-287	15	3	k=1	k=1	PUNCT
ma-287	16	1	(	(	PUNCT
ma-287	16	2	−1)k−1	−1)k−1	INTJ
ma-287	16	3	k	k	X
ma-287	16	4	(	(	PUNCT
ma-287	16	5	x	x	X
ma-287	16	6	−	−	PROPN
ma-287	16	7	1)k	1)k	NUM
ma-287	16	8	.	.	PUNCT
ma-287	17	1	it	it	PRON
ma-287	17	2	is	be	AUX
ma-287	17	3	valid	valid	ADJ
ma-287	17	4	for	for	SCONJ
ma-287	17	5	x	x	PROPN
ma-287	17	6	∈	∈	PROPN
ma-287	17	7	(	(	PUNCT
ma-287	17	8	0	0	NUM
ma-287	17	9	,	,	PUNCT
ma-287	17	10	2	2	NUM
ma-287	17	11	]	]	PUNCT
ma-287	17	12	only	only	ADV
ma-287	17	13	(	(	PUNCT
ma-287	17	14	see	see	VERB
ma-287	17	15	[	[	X
ma-287	17	16	8	8	NUM
ma-287	17	17	]	]	PUNCT
ma-287	17	18	,	,	PUNCT
ma-287	17	19	among	among	ADP
ma-287	17	20	others	other	NOUN
ma-287	17	21	)	)	PUNCT
ma-287	17	22	.	.	PUNCT
ma-287	18	1	from	from	ADP
ma-287	18	2	this	this	DET
ma-287	18	3	expansion	expansion	NOUN
ma-287	18	4	,	,	PUNCT
ma-287	18	5	we	we	PRON
ma-287	18	6	immediately	immediately	ADV
ma-287	18	7	get	get	VERB
ma-287	18	8	x	x	X
ma-287	18	9	−	−	PROPN
ma-287	18	10	1−	1−	NUM
ma-287	18	11	log(x	log(x	NUM
ma-287	18	12	)	)	PUNCT
ma-287	18	13	=	=	PUNCT
ma-287	19	1	+	+	ADJ
ma-287	19	2	∞∑	∞∑	ADJ
ma-287	19	3	k=2	k=2	PROPN
ma-287	19	4	1	1	NUM
ma-287	19	5	k	k	X
ma-287	19	6	(	(	PUNCT
ma-287	19	7	1−	1−	NUM
ma-287	19	8	x)k	x)k	NOUN
ma-287	19	9	.	.	PUNCT
ma-287	20	1	(	(	PUNCT
ma-287	20	2	1	1	X
ma-287	20	3	)	)	PUNCT
ma-287	20	4	a	a	DET
ma-287	20	5	consequence	consequence	NOUN
ma-287	20	6	of	of	ADP
ma-287	20	7	this	this	DET
ma-287	20	8	result	result	NOUN
ma-287	20	9	is	be	AUX
ma-287	20	10	the	the	DET
ma-287	20	11	following	follow	VERB
ma-287	20	12	inequality	inequality	NOUN
ma-287	20	13	:	:	PUNCT
ma-287	20	14	log(x	log(x	NUM
ma-287	20	15	)	)	PUNCT
ma-287	20	16	≤	≤	NOUN
ma-287	20	17	x	x	PUNCT
ma-287	21	1	−	−	PROPN
ma-287	21	2	1	1	NUM
ma-287	21	3	for	for	ADP
ma-287	21	4	x	x	PROPN
ma-287	21	5	∈	∈	PROPN
ma-287	21	6	(	(	PUNCT
ma-287	21	7	0	0	NUM
ma-287	21	8	,	,	PUNCT
ma-287	21	9	2	2	NUM
ma-287	21	10	]	]	PUNCT
ma-287	21	11	.	.	PUNCT
ma-287	22	1	however	however	ADV
ma-287	22	2	,	,	PUNCT
ma-287	22	3	asdiscussed	asdiscusse	VERB
ma-287	22	4	in	in	ADP
ma-287	22	5	[	[	X
ma-287	22	6	4	4	NUM
ma-287	22	7	]	]	PUNCT
ma-287	22	8	,	,	PUNCT
ma-287	22	9	we	we	PRON
ma-287	22	10	know	know	VERB
ma-287	22	11	that	that	SCONJ
ma-287	22	12	it	it	PRON
ma-287	22	13	actually	actually	ADV
ma-287	22	14	holds	hold	VERB
ma-287	22	15	for	for	ADP
ma-287	22	16	x	x	PUNCT
ma-287	22	17	>	>	X
ma-287	22	18	0	0	NUM
ma-287	22	19	.	.	PUNCT
ma-287	23	1	thus	thus	ADV
ma-287	23	2	,	,	PUNCT
ma-287	23	3	the	the	DET
ma-287	23	4	use	use	NOUN
ma-287	23	5	of	of	ADP
ma-287	23	6	the	the	DET
ma-287	23	7	classical	classical	ADJ
ma-287	23	8	logarithmicexpansion	logarithmicexpansion	NOUN
ma-287	23	9	is	be	AUX
ma-287	23	10	somehow	somehow	ADV
ma-287	23	11	inadequate	inadequate	ADJ
ma-287	23	12	for	for	ADP
ma-287	23	13	a	a	DET
ma-287	23	14	full	full	ADJ
ma-287	23	15	understanding	understanding	NOUN
ma-287	23	16	of	of	ADP
ma-287	23	17	this	this	DET
ma-287	23	18	inequality	inequality	NOUN
ma-287	23	19	.	.	PUNCT
ma-287	24	1	there	there	PRON
ma-287	24	2	is	be	VERB
ma-287	24	3	a	a	DET
ma-287	24	4	kind	kind	NOUN
ma-287	24	5	received	receive	VERB
ma-287	24	6	:	:	PUNCT
ma-287	24	7	26	26	NUM
ma-287	24	8	sep	sep	NOUN
ma-287	24	9	2024	2024	NUM
ma-287	24	10	.	.	PUNCT
ma-287	25	1	key	key	ADJ
ma-287	25	2	words	word	NOUN
ma-287	25	3	and	and	CCONJ
ma-287	25	4	phrases	phrase	NOUN
ma-287	25	5	.	.	PUNCT
ma-287	26	1	series	series	NOUN
ma-287	26	2	expansions	expansion	NOUN
ma-287	26	3	;	;	PUNCT
ma-287	26	4	logarithmic	logarithmic	ADJ
ma-287	26	5	function	function	NOUN
ma-287	26	6	;	;	PUNCT
ma-287	26	7	product	product	NOUN
ma-287	26	8	expansions	expansion	NOUN
ma-287	26	9	;	;	PUNCT
ma-287	26	10	inequalities.1	inequalities.1	ADP
ma-287	26	11	https://adac.ee	https://adac.ee	PROPN
ma-287	26	12	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	26	13	https://orcid.org/0000-0002-1522-9292	https://orcid.org/0000-0002-1522-9292	PROPN
ma-287	26	14	eur	eur	NOUN
ma-287	26	15	.	.	PUNCT
ma-287	27	1	j.	j.	PROPN
ma-287	27	2	math	math	PROPN
ma-287	27	3	.	.	PUNCT
ma-287	28	1	anal	anal	PROPN
ma-287	28	2	.	.	PUNCT
ma-287	29	1	10.28924	10.28924	NUM
ma-287	29	2	/	/	SYM
ma-287	29	3	ada	ada	PROPN
ma-287	29	4	/	/	SYM
ma-287	29	5	ma.5.12	ma.5.12	PROPN
ma-287	29	6	2of	2of	NOUN
ma-287	29	7	"	"	PUNCT
ma-287	29	8	gap	gap	NOUN
ma-287	29	9	in	in	ADP
ma-287	29	10	understanding	understanding	NOUN
ma-287	29	11	"	"	PUNCT
ma-287	29	12	.	.	PUNCT
ma-287	30	1	a	a	DET
ma-287	30	2	solution	solution	NOUN
ma-287	30	3	is	be	AUX
ma-287	30	4	given	give	VERB
ma-287	30	5	in	in	ADP
ma-287	30	6	[	[	X
ma-287	30	7	4	4	NUM
ma-287	30	8	]	]	PUNCT
ma-287	30	9	.	.	PUNCT
ma-287	31	1	using	use	VERB
ma-287	31	2	a	a	DET
ma-287	31	3	telescoping	telescope	VERB
ma-287	31	4	technique	technique	NOUN
ma-287	31	5	and	and	CCONJ
ma-287	31	6	carefuldevelopment	carefuldevelopment	NOUN
ma-287	31	7	,	,	PUNCT
ma-287	31	8	the	the	DET
ma-287	31	9	following	follow	VERB
ma-287	31	10	series	series	NOUN
ma-287	31	11	expansion	expansion	NOUN
ma-287	31	12	is	be	AUX
ma-287	31	13	demonstrated	demonstrate	VERB
ma-287	31	14	:	:	PUNCT
ma-287	31	15	x	x	X
ma-287	31	16	−	−	PROPN
ma-287	31	17	1−	1−	NUM
ma-287	31	18	log(x	log(x	NUM
ma-287	31	19	)	)	PUNCT
ma-287	31	20	=	=	PUNCT
ma-287	32	1	+	+	PUNCT
ma-287	32	2	∞∑	∞∑	NUM
ma-287	32	3	k=1	k=1	ADJ
ma-287	32	4	2k−1(x2	2k−1(x2	NOUN
ma-287	32	5	−k	−k	NOUN
ma-287	32	6	−	−	PROPN
ma-287	32	7	1)2	1)2	NUM
ma-287	32	8	,	,	PUNCT
ma-287	32	9	and	and	CCONJ
ma-287	32	10	it	it	PRON
ma-287	32	11	is	be	AUX
ma-287	32	12	valid	valid	ADJ
ma-287	32	13	for	for	ADP
ma-287	32	14	x	x	PUNCT
ma-287	32	15	>	>	X
ma-287	32	16	0	0	NUM
ma-287	32	17	.	.	PUNCT
ma-287	33	1	from	from	ADP
ma-287	33	2	this	this	DET
ma-287	33	3	result	result	NOUN
ma-287	33	4	,	,	PUNCT
ma-287	33	5	we	we	PRON
ma-287	33	6	immediately	immediately	ADV
ma-287	33	7	observe	observe	VERB
ma-287	33	8	that	that	SCONJ
ma-287	33	9	log(x	log(x	NOUN
ma-287	33	10	)	)	PUNCT
ma-287	33	11	≤	≤	NOUN
ma-287	33	12	x−1	x−1	PROPN
ma-287	34	1	for	for	ADP
ma-287	34	2	x	x	SYM
ma-287	34	3	>	>	X
ma-287	34	4	0	0	NUM
ma-287	34	5	;	;	PUNCT
ma-287	34	6	theconstraint	theconstraint	NOUN
ma-287	34	7	x	x	SYM
ma-287	34	8	∈	∈	PROPN
ma-287	34	9	(	(	PUNCT
ma-287	34	10	0	0	NUM
ma-287	34	11	,	,	PUNCT
ma-287	34	12	2	2	NUM
ma-287	34	13	]	]	PUNCT
ma-287	34	14	is	be	AUX
ma-287	34	15	relaxed	relax	VERB
ma-287	34	16	.	.	PUNCT
ma-287	35	1	this	this	DET
ma-287	35	2	key	key	ADJ
ma-287	35	3	inequality	inequality	NOUN
ma-287	35	4	is	be	AUX
ma-287	35	5	now	now	ADV
ma-287	35	6	fully	fully	ADV
ma-287	35	7	understandable	understandable	ADJ
ma-287	35	8	using	use	VERB
ma-287	35	9	the	the	DET
ma-287	35	10	seriesexpansion	seriesexpansion	NOUN
ma-287	35	11	tool	tool	NOUN
ma-287	35	12	.	.	PUNCT
ma-287	36	1	in	in	ADP
ma-287	36	2	addition	addition	NOUN
ma-287	36	3	,	,	PUNCT
ma-287	36	4	the	the	DET
ma-287	36	5	underlying	underlie	VERB
ma-287	36	6	telescoping	telescope	VERB
ma-287	36	7	technique	technique	NOUN
ma-287	36	8	provides	provide	VERB
ma-287	36	9	an	an	DET
ma-287	36	10	original	original	ADJ
ma-287	36	11	alternativeproof	alternativeproof	NOUN
ma-287	36	12	,	,	PUNCT
ma-287	36	13	making	make	VERB
ma-287	36	14	it	it	PRON
ma-287	36	15	very	very	ADV
ma-287	36	16	interesting	interesting	ADJ
ma-287	36	17	from	from	ADP
ma-287	36	18	a	a	DET
ma-287	36	19	mathematical	mathematical	ADJ
ma-287	36	20	point	point	NOUN
ma-287	36	21	of	of	ADP
ma-287	36	22	view	view	NOUN
ma-287	36	23	.	.	PUNCT
ma-287	37	1	it	it	PRON
ma-287	37	2	can	can	AUX
ma-287	37	3	also	also	ADV
ma-287	37	4	be	be	AUX
ma-287	37	5	used	use	VERB
ma-287	37	6	for	for	ADP
ma-287	37	7	otherpurposes	otherpurpose	NOUN
ma-287	37	8	;	;	PUNCT
ma-287	37	9	the	the	DET
ma-287	37	10	proof	proof	NOUN
ma-287	37	11	of	of	ADP
ma-287	37	12	natural	natural	ADJ
ma-287	37	13	logarithmic	logarithmic	ADJ
ma-287	37	14	inequalities	inequality	NOUN
ma-287	37	15	is	be	AUX
ma-287	37	16	just	just	ADV
ma-287	37	17	one	one	NUM
ma-287	37	18	example.on	example.on	NOUN
ma-287	37	19	the	the	DET
ma-287	37	20	other	other	ADJ
ma-287	37	21	hand	hand	NOUN
ma-287	37	22	,	,	PUNCT
ma-287	37	23	from	from	ADP
ma-287	37	24	a	a	DET
ma-287	37	25	completely	completely	ADV
ma-287	37	26	different	different	ADJ
ma-287	37	27	perspective	perspective	NOUN
ma-287	37	28	,	,	PUNCT
ma-287	37	29	a	a	DET
ma-287	37	30	famous	famous	ADJ
ma-287	37	31	result	result	NOUN
ma-287	37	32	of	of	ADP
ma-287	37	33	srinivasa	srinivasa	PROPN
ma-287	37	34	ramanu	ramanu	PROPN
ma-287	37	35	-	-	PUNCT
ma-287	37	36	jan	jan	PROPN
ma-287	37	37	ensures	ensure	VERB
ma-287	37	38	that	that	SCONJ
ma-287	37	39	,	,	PUNCT
ma-287	37	40	for	for	ADP
ma-287	37	41	x	x	X
ma-287	37	42	>	>	X
ma-287	37	43	0	0	AUX
ma-287	37	44	with	with	ADP
ma-287	37	45	x	x	SYM
ma-287	37	46	6=	6=	ADP
ma-287	37	47	1	1	NUM
ma-287	37	48	,	,	PUNCT
ma-287	37	49	we	we	PRON
ma-287	37	50	have	have	VERB
ma-287	37	51	1	1	NUM
ma-287	37	52	log(x	log(x	NUM
ma-287	37	53	)	)	PUNCT
ma-287	38	1	+	+	CCONJ
ma-287	38	2	1	1	NUM
ma-287	38	3	1−	1−	NUM
ma-287	38	4	x	x	X
ma-287	39	1	=	=	PUNCT
ma-287	39	2	+	+	ADP
ma-287	39	3	∞∑	∞∑	ADJ
ma-287	39	4	k=1	k=1	ADP
ma-287	39	5	1	1	NUM
ma-287	39	6	2k(1	2k(1	NOUN
ma-287	39	7	+	+	CCONJ
ma-287	39	8	x2	x2	PROPN
ma-287	39	9	−k	−k	PROPN
ma-287	39	10	)	)	PUNCT
ma-287	39	11	.	.	PUNCT
ma-287	40	1	see	see	VERB
ma-287	40	2	[	[	X
ma-287	40	3	12	12	NUM
ma-287	40	4	,	,	PUNCT
ma-287	40	5	page	page	NOUN
ma-287	40	6	364	364	NUM
ma-287	40	7	]	]	PUNCT
ma-287	40	8	.	.	PUNCT
ma-287	41	1	the	the	DET
ma-287	41	2	proof	proof	NOUN
ma-287	41	3	is	be	AUX
ma-287	41	4	based	base	VERB
ma-287	41	5	on	on	ADP
ma-287	41	6	an	an	DET
ma-287	41	7	iteration	iteration	NOUN
ma-287	41	8	technique	technique	NOUN
ma-287	41	9	,	,	PUNCT
ma-287	41	10	noting	note	VERB
ma-287	41	11	that	that	SCONJ
ma-287	41	12	1/(1	1/(1	NUM
ma-287	41	13	−	−	NOUN
ma-287	41	14	x	x	NOUN
ma-287	41	15	)	)	PUNCT
ma-287	42	1	=	=	SYM
ma-287	43	1	(	(	PUNCT
ma-287	43	2	1/2)[1/(1	1/2)[1/(1	NOUN
ma-287	43	3	+	+	CCONJ
ma-287	43	4	√	√	NUM
ma-287	43	5	x	x	NOUN
ma-287	43	6	)	)	PUNCT
ma-287	44	1	+	+	CCONJ
ma-287	45	1	1/(1	1/(1	NUM
ma-287	45	2	−	−	NOUN
ma-287	45	3	√	√	NUM
ma-287	45	4	x	x	SYM
ma-287	45	5	)	)	PUNCT
ma-287	45	6	]	]	PUNCT
ma-287	45	7	.	.	PUNCT
ma-287	46	1	it	it	PRON
ma-287	46	2	received	receive	VERB
ma-287	46	3	special	special	ADJ
ma-287	46	4	attention	attention	NOUN
ma-287	46	5	in	in	ADP
ma-287	46	6	[	[	X
ma-287	46	7	2	2	NUM
ma-287	46	8	,	,	PUNCT
ma-287	46	9	chapter	chapter	NOUN
ma-287	46	10	31	31	NUM
ma-287	46	11	,	,	PUNCT
ma-287	46	12	entry	entry	NOUN
ma-287	46	13	29	29	NUM
ma-287	46	14	,	,	PUNCT
ma-287	46	15	page399	page399	PROPN
ma-287	46	16	]	]	PUNCT
ma-287	46	17	and	and	CCONJ
ma-287	46	18	was	be	AUX
ma-287	46	19	the	the	DET
ma-287	46	20	object	object	NOUN
ma-287	46	21	of	of	ADP
ma-287	46	22	an	an	DET
ma-287	46	23	in	in	ADP
ma-287	46	24	-	-	PUNCT
ma-287	46	25	depth	depth	NOUN
ma-287	46	26	study	study	NOUN
ma-287	46	27	in	in	ADP
ma-287	46	28	[	[	X
ma-287	46	29	5	5	NUM
ma-287	46	30	]	]	PUNCT
ma-287	46	31	.	.	PUNCT
ma-287	47	1	as	as	ADP
ma-287	47	2	a	a	DET
ma-287	47	3	new	new	ADJ
ma-287	47	4	visual	visual	ADJ
ma-287	47	5	note	note	NOUN
ma-287	47	6	,	,	PUNCT
ma-287	47	7	this	this	DET
ma-287	47	8	extension	extension	NOUN
ma-287	47	9	can	can	AUX
ma-287	47	10	bereformulated	bereformulate	VERB
ma-287	47	11	as	as	ADP
ma-287	47	12	1	1	NUM
ma-287	47	13	x	x	SYM
ma-287	47	14	−	−	PROPN
ma-287	47	15	1	1	NUM
ma-287	47	16	−	−	PROPN
ma-287	47	17	1	1	NUM
ma-287	47	18	log(x	log(x	NOUN
ma-287	47	19	)	)	PUNCT
ma-287	47	20	=	=	PUNCT
ma-287	48	1	−	−	PUNCT
ma-287	49	1	+	+	NOUN
ma-287	49	2	∞∑	∞∑	ADJ
ma-287	49	3	k=1	k=1	ADP
ma-287	49	4	1	1	NUM
ma-287	49	5	2k(1	2k(1	NOUN
ma-287	49	6	+	+	CCONJ
ma-287	49	7	x2	x2	PROPN
ma-287	49	8	−k	−k	PROPN
ma-287	49	9	)	)	PUNCT
ma-287	49	10	.	.	PUNCT
ma-287	50	1	based	base	VERB
ma-287	50	2	on	on	ADP
ma-287	50	3	this	this	DET
ma-287	50	4	form	form	NOUN
ma-287	50	5	,	,	PUNCT
ma-287	50	6	doing	do	VERB
ma-287	50	7	a	a	DET
ma-287	50	8	parallel	parallel	NOUN
ma-287	50	9	with	with	ADP
ma-287	50	10	the	the	DET
ma-287	50	11	formula	formula	NOUN
ma-287	50	12	in	in	ADP
ma-287	50	13	equation	equation	NOUN
ma-287	50	14	(	(	PUNCT
ma-287	50	15	1	1	NUM
ma-287	50	16	)	)	PUNCT
ma-287	50	17	,	,	PUNCT
ma-287	50	18	a	a	DET
ma-287	50	19	functional	functional	ADJ
ma-287	50	20	pattern	pattern	NOUN
ma-287	50	21	seemsto	seemsto	NOUN
ma-287	50	22	be	be	AUX
ma-287	50	23	present	present	ADJ
ma-287	50	24	.	.	PUNCT
ma-287	51	1	in	in	ADP
ma-287	51	2	fact	fact	NOUN
ma-287	51	3	,	,	PUNCT
ma-287	51	4	both	both	DET
ma-287	51	5	expansions	expansion	NOUN
ma-287	51	6	can	can	AUX
ma-287	51	7	be	be	AUX
ma-287	51	8	expressed	express	VERB
ma-287	51	9	as	as	ADP
ma-287	51	10	φ(x	φ(x	PROPN
ma-287	51	11	−	−	PROPN
ma-287	51	12	1)−	1)−	PROPN
ma-287	51	13	φ[log(x	φ[log(x	NUM
ma-287	51	14	)	)	PUNCT
ma-287	51	15	]	]	PUNCT
ma-287	52	1	=	=	PUNCT
ma-287	53	1	+	+	PUNCT
ma-287	53	2	∞∑	∞∑	NOUN
ma-287	53	3	k=1	k=1	ADJ
ma-287	53	4	ak(x	ak(x	NOUN
ma-287	53	5	)	)	PUNCT
ma-287	53	6	,	,	PUNCT
ma-287	53	7	(	(	PUNCT
ma-287	53	8	2	2	X
ma-287	53	9	)	)	PUNCT
ma-287	53	10	where	where	SCONJ
ma-287	53	11	φ(t	φ(t	NOUN
ma-287	53	12	)	)	PUNCT
ma-287	53	13	=	=	SYM
ma-287	53	14	t	t	PROPN
ma-287	53	15	and	and	CCONJ
ma-287	53	16	φ(t	φ(t	PROPN
ma-287	53	17	)	)	PUNCT
ma-287	53	18	=	=	SYM
ma-287	53	19	1	1	NUM
ma-287	53	20	/	/	SYM
ma-287	53	21	t	t	NOUN
ma-287	53	22	,	,	PUNCT
ma-287	53	23	respectively	respectively	ADV
ma-287	53	24	,	,	PUNCT
ma-287	53	25	and	and	CCONJ
ma-287	53	26	ak(x	ak(x	NUM
ma-287	53	27	)	)	PUNCT
ma-287	53	28	=	=	SYM
ma-287	53	29	2k−1(x2−k−1)2	2k−1(x2−k−1)2	NUM
ma-287	53	30	and	and	CCONJ
ma-287	53	31	ak(x	ak(x	NUM
ma-287	53	32	)	)	PUNCT
ma-287	54	1	=	=	SYM
ma-287	54	2	−1/2k(1	−1/2k(1	PROPN
ma-287	54	3	+	+	CCONJ
ma-287	54	4	x2	x2	NOUN
ma-287	54	5	−k	−k	PROPN
ma-287	54	6	)	)	PUNCT
ma-287	54	7	,	,	PUNCT
ma-287	54	8	respectively	respectively	ADV
ma-287	54	9	.	.	PUNCT
ma-287	55	1	given	give	VERB
ma-287	55	2	this	this	PRON
ma-287	55	3	,	,	PUNCT
ma-287	55	4	a	a	DET
ma-287	55	5	unified	unified	ADJ
ma-287	55	6	approach	approach	NOUN
ma-287	55	7	seems	seem	VERB
ma-287	55	8	possible.in	possible.in	X
ma-287	55	9	this	this	DET
ma-287	55	10	article	article	NOUN
ma-287	55	11	,	,	PUNCT
ma-287	55	12	we	we	PRON
ma-287	55	13	formalize	formalize	VERB
ma-287	55	14	such	such	DET
ma-287	55	15	an	an	DET
ma-287	55	16	approach	approach	NOUN
ma-287	55	17	.	.	PUNCT
ma-287	56	1	it	it	PRON
ma-287	56	2	aims	aim	VERB
ma-287	56	3	to	to	PART
ma-287	56	4	generate	generate	VERB
ma-287	56	5	a	a	DET
ma-287	56	6	wide	wide	ADJ
ma-287	56	7	range	range	NOUN
ma-287	56	8	of	of	ADP
ma-287	56	9	new	new	ADJ
ma-287	56	10	seriesexpansions	seriesexpansion	NOUN
ma-287	56	11	of	of	ADP
ma-287	56	12	certain	certain	ADJ
ma-287	56	13	functions	function	NOUN
ma-287	56	14	that	that	PRON
ma-287	56	15	depend	depend	VERB
ma-287	56	16	on	on	ADP
ma-287	56	17	the	the	DET
ma-287	56	18	logarithmic	logarithmic	ADJ
ma-287	56	19	function	function	NOUN
ma-287	56	20	,	,	PUNCT
ma-287	56	21	i.e.	i.e.	X
ma-287	56	22	,	,	PUNCT
ma-287	56	23	functions	function	NOUN
ma-287	56	24	of	of	ADP
ma-287	56	25	the	the	DET
ma-287	56	26	form	form	NOUN
ma-287	56	27	φ(x	φ(x	PROPN
ma-287	56	28	−	−	PROPN
ma-287	56	29	1	1	NUM
ma-287	56	30	)	)	PUNCT
ma-287	56	31	−	−	PROPN
ma-287	56	32	φ[log(x	φ[log(x	NOUN
ma-287	56	33	)	)	PUNCT
ma-287	56	34	]	]	PUNCT
ma-287	56	35	.	.	PUNCT
ma-287	57	1	the	the	DET
ma-287	57	2	proof	proof	NOUN
ma-287	57	3	is	be	AUX
ma-287	57	4	based	base	VERB
ma-287	57	5	on	on	ADP
ma-287	57	6	telescoping	telescope	VERB
ma-287	57	7	techniques	technique	NOUN
ma-287	57	8	inspired	inspire	VERB
ma-287	57	9	by	by	ADP
ma-287	57	10	[	[	X
ma-287	57	11	4	4	NUM
ma-287	57	12	]	]	PUNCT
ma-287	57	13	and	and	CCONJ
ma-287	57	14	precisefactorization	precisefactorization	NOUN
ma-287	57	15	developments	development	NOUN
ma-287	57	16	.	.	PUNCT
ma-287	58	1	the	the	DET
ma-287	58	2	results	result	NOUN
ma-287	58	3	established	establish	VERB
ma-287	58	4	have	have	VERB
ma-287	58	5	interesting	interesting	ADJ
ma-287	58	6	consequences	consequence	NOUN
ma-287	58	7	,	,	PUNCT
ma-287	58	8	including	include	VERB
ma-287	58	9	thederivation	thederivation	NOUN
ma-287	58	10	of	of	ADP
ma-287	58	11	old	old	ADJ
ma-287	58	12	and	and	CCONJ
ma-287	58	13	new	new	ADJ
ma-287	58	14	product	product	NOUN
ma-287	58	15	expansions	expansion	NOUN
ma-287	58	16	.	.	PUNCT
ma-287	59	1	among	among	ADP
ma-287	59	2	other	other	ADJ
ma-287	59	3	things	thing	NOUN
ma-287	59	4	,	,	PUNCT
ma-287	59	5	a	a	DET
ma-287	59	6	new	new	ADJ
ma-287	59	7	product	product	NOUN
ma-287	59	8	expansion	expansion	NOUN
ma-287	59	9	ofthe	ofthe	PROPN
ma-287	59	10	einstein	einstein	PROPN
ma-287	59	11	function	function	PROPN
ma-287	59	12	,	,	PUNCT
ma-287	59	13	i.e.	i.e.	X
ma-287	59	14	,	,	PUNCT
ma-287	59	15	e2(x	e2(x	NOUN
ma-287	59	16	)	)	PUNCT
ma-287	59	17	=	=	PUNCT
ma-287	60	1	x/(ex	x/(ex	PROPN
ma-287	60	2	−	−	NOUN
ma-287	61	1	1	1	NUM
ma-287	61	2	)	)	PUNCT
ma-287	61	3	(	(	PUNCT
ma-287	61	4	see	see	VERB
ma-287	61	5	[	[	X
ma-287	61	6	1	1	NUM
ma-287	61	7	]	]	NUM
ma-287	61	8	)	)	PUNCT
ma-287	61	9	,	,	PUNCT
ma-287	61	10	is	be	AUX
ma-287	61	11	established	establish	VERB
ma-287	61	12	.	.	PUNCT
ma-287	62	1	in	in	ADP
ma-287	62	2	addition	addition	NOUN
ma-287	62	3	,	,	PUNCT
ma-287	62	4	inequalitiesinvolving	inequalitiesinvolve	VERB
ma-287	62	5	logarithmic	logarithmic	ADJ
ma-287	62	6	functions	function	NOUN
ma-287	62	7	are	be	AUX
ma-287	62	8	obtained	obtain	VERB
ma-287	62	9	almost	almost	ADV
ma-287	62	10	immediately	immediately	ADV
ma-287	62	11	,	,	PUNCT
ma-287	62	12	in	in	ADP
ma-287	62	13	the	the	DET
ma-287	62	14	spirit	spirit	NOUN
ma-287	62	15	of	of	ADP
ma-287	62	16	[	[	X
ma-287	62	17	3	3	NUM
ma-287	62	18	,	,	PUNCT
ma-287	62	19	6	6	NUM
ma-287	62	20	,	,	PUNCT
ma-287	62	21	7	7	NUM
ma-287	62	22	,	,	PUNCT
ma-287	62	23	9–11	9–11	NOUN
ma-287	62	24	,	,	PUNCT
ma-287	62	25	16].some	16].some	PROPN
ma-287	62	26	graphics	graphic	NOUN
ma-287	62	27	illustrate	illustrate	VERB
ma-287	62	28	the	the	DET
ma-287	62	29	results.the	results.the	DET
ma-287	62	30	following	follow	VERB
ma-287	62	31	sections	section	NOUN
ma-287	62	32	structure	structure	VERB
ma-287	62	33	the	the	DET
ma-287	62	34	article	article	NOUN
ma-287	62	35	:	:	PUNCT
ma-287	62	36	section	section	NOUN
ma-287	62	37	2	2	NUM
ma-287	62	38	is	be	AUX
ma-287	62	39	devoted	devote	VERB
ma-287	62	40	to	to	ADP
ma-287	62	41	the	the	DET
ma-287	62	42	general	general	ADJ
ma-287	62	43	result	result	NOUN
ma-287	62	44	on	on	ADP
ma-287	62	45	theseries	theserie	NOUN
ma-287	62	46	expansion	expansion	NOUN
ma-287	62	47	of	of	ADP
ma-287	62	48	φ(x	φ(x	PROPN
ma-287	62	49	−	−	PROPN
ma-287	62	50	1	1	NUM
ma-287	62	51	)	)	PUNCT
ma-287	62	52	−	−	PROPN
ma-287	63	1	φ[log(x	φ[log(x	NOUN
ma-287	64	1	)	)	PUNCT
ma-287	65	1	]	]	PUNCT
ma-287	66	1	and	and	CCONJ
ma-287	66	2	emphasizes	emphasize	VERB
ma-287	66	3	several	several	ADJ
ma-287	66	4	examples	example	NOUN
ma-287	66	5	.	.	PUNCT
ma-287	67	1	section	section	NOUN
ma-287	67	2	3	3	NUM
ma-287	67	3	deals	deal	NOUN
ma-287	67	4	with	with	ADP
ma-287	67	5	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	67	6	eur	eur	PROPN
ma-287	67	7	.	.	PUNCT
ma-287	68	1	j.	j.	PROPN
ma-287	68	2	math	math	PROPN
ma-287	68	3	.	.	PUNCT
ma-287	69	1	anal	anal	PROPN
ma-287	69	2	.	.	PUNCT
ma-287	70	1	10.28924	10.28924	NUM
ma-287	70	2	/	/	SYM
ma-287	70	3	ada	ada	PROPN
ma-287	70	4	/	/	SYM
ma-287	70	5	ma.5.12	ma.5.12	PROPN
ma-287	70	6	3some	3some	NUM
ma-287	70	7	of	of	ADP
ma-287	70	8	its	its	PRON
ma-287	70	9	implications	implication	NOUN
ma-287	70	10	,	,	PUNCT
ma-287	70	11	including	include	VERB
ma-287	70	12	product	product	NOUN
ma-287	70	13	expansions	expansion	NOUN
ma-287	70	14	and	and	CCONJ
ma-287	70	15	inequalities	inequality	NOUN
ma-287	70	16	.	.	PUNCT
ma-287	71	1	a	a	DET
ma-287	71	2	conclusion	conclusion	NOUN
ma-287	71	3	is	be	AUX
ma-287	71	4	given	give	VERB
ma-287	71	5	insection	insection	NOUN
ma-287	71	6	4	4	NUM
ma-287	71	7	.	.	NOUN
ma-287	71	8	2	2	NUM
ma-287	71	9	.	.	X
ma-287	71	10	results	result	VERB
ma-287	71	11	our	our	PRON
ma-287	71	12	general	general	ADJ
ma-287	71	13	and	and	CCONJ
ma-287	71	14	specific	specific	ADJ
ma-287	71	15	findings	finding	NOUN
ma-287	71	16	are	be	AUX
ma-287	71	17	presented	present	VERB
ma-287	71	18	in	in	ADP
ma-287	71	19	this	this	DET
ma-287	71	20	section	section	NOUN
ma-287	71	21	.	.	PUNCT
ma-287	72	1	2.1	2.1	NUM
ma-287	72	2	.	.	PUNCT
ma-287	73	1	a	a	DET
ma-287	73	2	general	general	ADJ
ma-287	73	3	result	result	NOUN
ma-287	73	4	.	.	PUNCT
ma-287	74	1	the	the	DET
ma-287	74	2	theorem	theorem	NOUN
ma-287	74	3	below	below	ADV
ma-287	74	4	suggests	suggest	VERB
ma-287	74	5	a	a	DET
ma-287	74	6	series	series	NOUN
ma-287	74	7	expansion	expansion	NOUN
ma-287	74	8	for	for	ADP
ma-287	74	9	the	the	DET
ma-287	74	10	difference	difference	NOUN
ma-287	74	11	function	function	NOUN
ma-287	74	12	φ(x	φ(x	PROPN
ma-287	74	13	−	−	PROPN
ma-287	74	14	1)−	1)−	PROPN
ma-287	74	15	φ[log(x	φ[log(x	NUM
ma-287	74	16	)	)	PUNCT
ma-287	74	17	]	]	PUNCT
ma-287	74	18	,	,	PUNCT
ma-287	74	19	defined	define	VERB
ma-287	74	20	with	with	ADP
ma-287	74	21	a	a	DET
ma-287	74	22	certain	certain	ADJ
ma-287	74	23	function	function	NOUN
ma-287	74	24	φ	φ	PROPN
ma-287	74	25	.	.	PUNCT
ma-287	74	26	theorem	theorem	VERB
ma-287	74	27	2.1	2.1	NUM
ma-287	74	28	.	.	PUNCT
ma-287	75	1	let	let	VERB
ma-287	75	2	x	x	PRON
ma-287	75	3	>	>	PUNCT
ma-287	75	4	0	0	PUNCT
ma-287	76	1	and	and	CCONJ
ma-287	76	2	φ	φ	PROPN
ma-287	76	3	be	be	VERB
ma-287	76	4	a	a	DET
ma-287	76	5	continuous	continuous	ADJ
ma-287	76	6	function	function	NOUN
ma-287	76	7	such	such	ADJ
ma-287	76	8	that	that	DET
ma-287	76	9	|φ(x−1)|	|φ(x−1)|	NOUN
ma-287	76	10	<	<	X
ma-287	77	1	+	+	NOUN
ma-287	77	2	∞	∞	NUM
ma-287	77	3	and	and	CCONJ
ma-287	77	4	|φ[log(x)]|	|φ[log(x)]|	X
ma-287	77	5	<	<	X
ma-287	77	6	+	+	X
ma-287	77	7	∞.	∞.	PROPN
ma-287	77	8	then	then	ADV
ma-287	77	9	we	we	PRON
ma-287	77	10	have	have	VERB
ma-287	77	11	φ(x	φ(x	PROPN
ma-287	77	12	−	−	PROPN
ma-287	77	13	1)−	1)−	PROPN
ma-287	77	14	φ[log(x	φ[log(x	NUM
ma-287	77	15	)	)	PUNCT
ma-287	77	16	]	]	PUNCT
ma-287	78	1	=	=	PUNCT
ma-287	79	1	+	+	PUNCT
ma-287	79	2	∞∑	∞∑	NUM
ma-287	79	3	k=1	k=1	ADJ
ma-287	79	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	79	5	)	)	PUNCT
ma-287	79	6	,	,	PUNCT
ma-287	79	7	where	where	SCONJ
ma-287	79	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	79	9	)	)	PUNCT
ma-287	80	1	=	=	PUNCT
ma-287	80	2	φ	φ	PROPN
ma-287	80	3	[	[	PUNCT
ma-287	80	4	2k−1(x2	2k−1(x2	NOUN
ma-287	80	5	−(k−1	−(k−1	NOUN
ma-287	80	6	)	)	PUNCT
ma-287	81	1	−	−	PROPN
ma-287	81	2	1	1	NUM
ma-287	81	3	)	)	PUNCT
ma-287	81	4	]	]	PUNCT
ma-287	82	1	−	−	PROPN
ma-287	82	2	φ	φ	PROPN
ma-287	82	3	[	[	PUNCT
ma-287	82	4	2k(x2	2k(x2	NOUN
ma-287	82	5	−k	−k	ADJ
ma-287	82	6	−	−	NOUN
ma-287	82	7	1	1	NUM
ma-287	82	8	)	)	PUNCT
ma-287	82	9	]	]	PUNCT
ma-287	82	10	.	.	PUNCT
ma-287	83	1	proof	proof	NOUN
ma-287	83	2	.	.	PUNCT
ma-287	84	1	introducing	introduce	VERB
ma-287	84	2	an	an	DET
ma-287	84	3	integer	integer	NOUN
ma-287	84	4	n	n	PRON
ma-287	84	5	≥	≥	NOUN
ma-287	84	6	1	1	NUM
ma-287	84	7	and	and	CCONJ
ma-287	84	8	using	use	VERB
ma-287	84	9	the	the	DET
ma-287	84	10	telescoping	telescope	VERB
ma-287	84	11	technique	technique	NOUN
ma-287	84	12	,	,	PUNCT
ma-287	84	13	we	we	PRON
ma-287	84	14	get	get	VERB
ma-287	84	15	φ(x	φ(x	PROPN
ma-287	84	16	−	−	PROPN
ma-287	84	17	1)−	1)−	NUM
ma-287	84	18	φ	φ	PROPN
ma-287	84	19	[	[	PUNCT
ma-287	84	20	2n(x2	2n(x2	NUM
ma-287	84	21	−n	−n	NOUN
ma-287	84	22	−	−	PROPN
ma-287	84	23	1	1	NUM
ma-287	84	24	)	)	PUNCT
ma-287	84	25	]	]	PUNCT
ma-287	85	1	=	=	PUNCT
ma-287	85	2	φ	φ	PROPN
ma-287	85	3	[	[	PUNCT
ma-287	85	4	20(x2	20(x2	NUM
ma-287	85	5	−0	−0	NOUN
ma-287	85	6	−	−	NOUN
ma-287	85	7	1	1	NUM
ma-287	85	8	)	)	PUNCT
ma-287	85	9	]	]	PUNCT
ma-287	86	1	−	−	PROPN
ma-287	86	2	φ	φ	X
ma-287	86	3	[	[	PUNCT
ma-287	86	4	2n(x2	2n(x2	NUM
ma-287	86	5	−n	−n	NOUN
ma-287	86	6	−	−	PROPN
ma-287	86	7	1	1	NUM
ma-287	86	8	)	)	PUNCT
ma-287	86	9	]	]	PUNCT
ma-287	87	1	=	=	PUNCT
ma-287	87	2	n∑	n∑	INTJ
ma-287	87	3	k=1	k=1	PROPN
ma-287	87	4	{	{	PUNCT
ma-287	87	5	φ	φ	X
ma-287	87	6	[	[	PUNCT
ma-287	87	7	2k−1(x2	2k−1(x2	NOUN
ma-287	87	8	−(k−1	−(k−1	NOUN
ma-287	87	9	)	)	PUNCT
ma-287	87	10	−	−	PROPN
ma-287	88	1	1	1	NUM
ma-287	88	2	)	)	PUNCT
ma-287	88	3	]	]	PUNCT
ma-287	89	1	−	−	PROPN
ma-287	89	2	φ	φ	PROPN
ma-287	89	3	[	[	PUNCT
ma-287	89	4	2k(x2	2k(x2	NOUN
ma-287	89	5	−k	−k	ADJ
ma-287	89	6	−	−	NOUN
ma-287	89	7	1	1	NUM
ma-287	89	8	)	)	PUNCT
ma-287	89	9	]	]	PUNCT
ma-287	89	10	}	}	PUNCT
ma-287	89	11	=	=	PUNCT
ma-287	89	12	n∑	n∑	NOUN
ma-287	89	13	k=1	k=1	PROPN
ma-287	89	14	αk(φ)(x	αk(φ)(x	PROPN
ma-287	89	15	)	)	PUNCT
ma-287	89	16	.	.	PUNCT
ma-287	90	1	since	since	SCONJ
ma-287	90	2	limn→+∞	limn→+∞	PROPN
ma-287	90	3	2n(x2−n−1	2n(x2−n−1	NUM
ma-287	90	4	)	)	PUNCT
ma-287	90	5	=	=	PRON
ma-287	90	6	limn→+∞	limn→+∞	VERB
ma-287	90	7	2n(e2−n	2n(e2−n	NUM
ma-287	90	8	log(x)−1	log(x)−1	NOUN
ma-287	90	9	)	)	PUNCT
ma-287	91	1	=	=	SYM
ma-287	91	2	limn→+∞	limn→+∞	VERB
ma-287	91	3	2n	2n	NUM
ma-287	91	4	{	{	PUNCT
ma-287	91	5	[	[	X
ma-287	91	6	1	1	NUM
ma-287	91	7	+	+	NUM
ma-287	91	8	2−n	2−n	NUM
ma-287	91	9	log(x)]−	log(x)]−	NOUN
ma-287	91	10	1	1	NUM
ma-287	91	11	}	}	PUNCT
ma-287	91	12	=	=	SYM
ma-287	91	13	log(x	log(x	PROPN
ma-287	91	14	)	)	PUNCT
ma-287	91	15	,	,	PUNCT
ma-287	91	16	thanks	thank	NOUN
ma-287	91	17	to	to	ADP
ma-287	91	18	the	the	DET
ma-287	91	19	continuity	continuity	NOUN
ma-287	91	20	of	of	ADP
ma-287	91	21	φ	φ	PROPN
ma-287	91	22	,	,	PUNCT
ma-287	91	23	we	we	PRON
ma-287	91	24	obtain	obtain	VERB
ma-287	91	25	φ(x	φ(x	PROPN
ma-287	91	26	−	−	PROPN
ma-287	91	27	1)−	1)−	NUM
ma-287	91	28	φ[log(x	φ[log(x	NUM
ma-287	91	29	)	)	PUNCT
ma-287	91	30	]	]	PUNCT
ma-287	92	1	=	=	PUNCT
ma-287	92	2	φ(x	φ(x	PROPN
ma-287	92	3	−	−	PROPN
ma-287	92	4	1)−	1)−	NUM
ma-287	92	5	φ	φ	PROPN
ma-287	92	6	[	[	PUNCT
ma-287	92	7	lim	lim	PROPN
ma-287	92	8	n→+∞	n→+∞	VERB
ma-287	92	9	2n(x2	2n(x2	NUM
ma-287	92	10	−n	−n	NOUN
ma-287	92	11	−	−	PROPN
ma-287	92	12	1	1	NUM
ma-287	92	13	)	)	PUNCT
ma-287	92	14	]	]	PUNCT
ma-287	93	1	=	=	SYM
ma-287	93	2	lim	lim	PROPN
ma-287	93	3	n→+∞	n→+∞	PROPN
ma-287	93	4	{	{	PUNCT
ma-287	93	5	φ(x	φ(x	PROPN
ma-287	93	6	−	−	PROPN
ma-287	94	1	1)−	1)−	NUM
ma-287	94	2	φ	φ	PROPN
ma-287	94	3	[	[	PUNCT
ma-287	94	4	2n(x2	2n(x2	NUM
ma-287	94	5	−n	−n	NOUN
ma-287	94	6	−	−	PROPN
ma-287	94	7	1	1	NUM
ma-287	94	8	)	)	PUNCT
ma-287	94	9	]	]	PUNCT
ma-287	94	10	}	}	PUNCT
ma-287	94	11	=	=	SYM
ma-287	94	12	lim	lim	PROPN
ma-287	94	13	n→+∞	n→+∞	VERB
ma-287	95	1	n∑	n∑	PROPN
ma-287	95	2	k=1	k=1	PROPN
ma-287	95	3	αk(φ)(x	αk(φ)(x	PROPN
ma-287	95	4	)	)	PUNCT
ma-287	96	1	=	=	PUNCT
ma-287	97	1	+	+	PUNCT
ma-287	97	2	∞∑	∞∑	NUM
ma-287	97	3	k=1	k=1	X
ma-287	97	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	97	5	)	)	PUNCT
ma-287	97	6	.	.	PUNCT
ma-287	98	1	this	this	PRON
ma-287	98	2	ends	end	VERB
ma-287	98	3	the	the	DET
ma-287	98	4	proof	proof	NOUN
ma-287	98	5	of	of	ADP
ma-287	98	6	the	the	DET
ma-287	98	7	theorem	theorem	PROPN
ma-287	98	8	.	.	PUNCT
ma-287	98	9	�	�	PROPN
ma-287	98	10	from	from	ADP
ma-287	98	11	this	this	DET
ma-287	98	12	theorem	theorem	NOUN
ma-287	98	13	,	,	PUNCT
ma-287	98	14	if	if	SCONJ
ma-287	98	15	φ	φ	PROPN
ma-287	98	16	is	be	AUX
ma-287	98	17	bijective	bijective	ADJ
ma-287	98	18	,	,	PUNCT
ma-287	98	19	then	then	ADV
ma-287	98	20	the	the	DET
ma-287	98	21	following	follow	VERB
ma-287	98	22	series	series	NOUN
ma-287	98	23	expansion	expansion	NOUN
ma-287	98	24	of	of	ADP
ma-287	98	25	the	the	DET
ma-287	98	26	logarithmic	logarithmic	ADJ
ma-287	98	27	functionholds	functionhold	NOUN
ma-287	98	28	:	:	PUNCT
ma-287	98	29	log(x	log(x	X
ma-287	98	30	)	)	PUNCT
ma-287	98	31	=	=	SYM
ma-287	98	32	φ−1	φ−1	PROPN
ma-287	98	33	{	{	PUNCT
ma-287	98	34	φ(x	φ(x	PROPN
ma-287	98	35	−	−	PROPN
ma-287	98	36	1)−	1)−	PROPN
ma-287	98	37	+	+	PROPN
ma-287	98	38	∞∑	∞∑	ADJ
ma-287	98	39	k=1	k=1	X
ma-287	98	40	αk(φ)(x	αk(φ)(x	PROPN
ma-287	98	41	)	)	PUNCT
ma-287	98	42	}	}	PUNCT
ma-287	98	43	.	.	PUNCT
ma-287	99	1	it	it	PRON
ma-287	99	2	can	can	AUX
ma-287	99	3	be	be	AUX
ma-287	99	4	useful	useful	ADJ
ma-287	99	5	in	in	ADP
ma-287	99	6	several	several	ADJ
ma-287	99	7	mathematical	mathematical	ADJ
ma-287	99	8	contexts	contexts	NOUN
ma-287	99	9	.	.	PUNCT
ma-287	100	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	100	2	eur	eur	PROPN
ma-287	100	3	.	.	PUNCT
ma-287	101	1	j.	j.	PROPN
ma-287	101	2	math	math	PROPN
ma-287	101	3	.	.	PUNCT
ma-287	102	1	anal	anal	PROPN
ma-287	102	2	.	.	PUNCT
ma-287	103	1	10.28924	10.28924	NUM
ma-287	103	2	/	/	SYM
ma-287	103	3	ada	ada	PROPN
ma-287	103	4	/	/	SYM
ma-287	103	5	ma.5.12	ma.5.12	PROPN
ma-287	103	6	4	4	NUM
ma-287	103	7	in	in	ADP
ma-287	103	8	fact	fact	NOUN
ma-287	103	9	,	,	PUNCT
ma-287	103	10	this	this	DET
ma-287	103	11	theorem	theorem	NOUN
ma-287	103	12	has	have	VERB
ma-287	103	13	the	the	DET
ma-287	103	14	advantage	advantage	NOUN
ma-287	103	15	of	of	ADP
ma-287	103	16	being	be	AUX
ma-287	103	17	general	general	ADJ
ma-287	103	18	and	and	CCONJ
ma-287	103	19	tunable	tunable	ADJ
ma-287	103	20	thanks	thank	NOUN
ma-287	103	21	to	to	ADP
ma-287	103	22	the	the	DET
ma-287	103	23	function	function	NOUN
ma-287	103	24	φ.it	φ.it	PROPN
ma-287	103	25	also	also	ADV
ma-287	103	26	unifies	unify	VERB
ma-287	103	27	several	several	ADJ
ma-287	103	28	results	result	NOUN
ma-287	103	29	in	in	ADP
ma-287	103	30	the	the	DET
ma-287	103	31	literature	literature	NOUN
ma-287	103	32	,	,	PUNCT
ma-287	103	33	as	as	SCONJ
ma-287	103	34	will	will	AUX
ma-287	103	35	be	be	AUX
ma-287	103	36	developed	develop	VERB
ma-287	103	37	in	in	ADP
ma-287	103	38	the	the	DET
ma-287	103	39	next	next	ADJ
ma-287	103	40	part	part	NOUN
ma-287	103	41	.	.	PUNCT
ma-287	104	1	2.2	2.2	NUM
ma-287	104	2	.	.	PUNCT
ma-287	104	3	specific	specific	ADJ
ma-287	104	4	results	result	NOUN
ma-287	104	5	.	.	PUNCT
ma-287	105	1	the	the	DET
ma-287	105	2	proposition	proposition	NOUN
ma-287	105	3	below	below	ADP
ma-287	105	4	shows	show	VERB
ma-287	105	5	some	some	DET
ma-287	105	6	consequences	consequence	NOUN
ma-287	105	7	of	of	ADP
ma-287	105	8	theorem	theorem	ADJ
ma-287	105	9	2.1	2.1	NUM
ma-287	105	10	,	,	PUNCT
ma-287	105	11	includingsome	includingsome	ADJ
ma-287	105	12	new	new	ADJ
ma-287	105	13	results	result	NOUN
ma-287	105	14	.	.	PUNCT
ma-287	106	1	proposition	proposition	NOUN
ma-287	106	2	2.2	2.2	NUM
ma-287	106	3	.	.	PUNCT
ma-287	107	1	the	the	DET
ma-287	107	2	series	series	NOUN
ma-287	107	3	expansions	expansion	NOUN
ma-287	107	4	below	below	ADV
ma-287	107	5	are	be	AUX
ma-287	107	6	valid.(1	valid.(1	PROPN
ma-287	107	7	)	)	PUNCT
ma-287	107	8	for	for	ADP
ma-287	107	9	x	x	SYM
ma-287	107	10	>	>	X
ma-287	107	11	0	0	NUM
ma-287	107	12	,	,	PUNCT
ma-287	107	13	we	we	PRON
ma-287	107	14	have	have	VERB
ma-287	107	15	x	x	X
ma-287	107	16	−	−	PROPN
ma-287	107	17	1−	1−	NUM
ma-287	107	18	log(x	log(x	NUM
ma-287	107	19	)	)	PUNCT
ma-287	107	20	=	=	PUNCT
ma-287	108	1	+	+	PUNCT
ma-287	108	2	∞∑	∞∑	NUM
ma-287	108	3	k=1	k=1	ADJ
ma-287	108	4	2k−1(x2	2k−1(x2	NOUN
ma-287	108	5	−k	−k	NOUN
ma-287	108	6	−	−	PROPN
ma-287	108	7	1)2	1)2	NUM
ma-287	108	8	.	.	PUNCT
ma-287	109	1	as	as	SCONJ
ma-287	109	2	mentioned	mention	VERB
ma-287	109	3	in	in	ADP
ma-287	109	4	the	the	DET
ma-287	109	5	introduction	introduction	NOUN
ma-287	109	6	,	,	PUNCT
ma-287	109	7	this	this	DET
ma-287	109	8	result	result	NOUN
ma-287	109	9	is	be	AUX
ma-287	109	10	not	not	PART
ma-287	109	11	new	new	ADJ
ma-287	109	12	;	;	PUNCT
ma-287	109	13	it	it	PRON
ma-287	109	14	was	be	AUX
ma-287	109	15	established	establish	VERB
ma-287	109	16	in	in	ADP
ma-287	109	17	[	[	X
ma-287	109	18	4].(2	4].(2	NUM
ma-287	109	19	)	)	PUNCT
ma-287	109	20	for	for	ADP
ma-287	109	21	x	x	SYM
ma-287	109	22	>	>	X
ma-287	109	23	0	0	NUM
ma-287	109	24	,	,	PUNCT
ma-287	109	25	we	we	PRON
ma-287	109	26	have	have	VERB
ma-287	109	27	1	1	NUM
ma-287	109	28	x	x	SYM
ma-287	109	29	−	−	NOUN
ma-287	109	30	1	1	NUM
ma-287	109	31	−	−	PROPN
ma-287	109	32	1	1	NUM
ma-287	109	33	log(x	log(x	NOUN
ma-287	109	34	)	)	PUNCT
ma-287	109	35	=	=	PUNCT
ma-287	110	1	−	−	PUNCT
ma-287	111	1	+	+	NOUN
ma-287	111	2	∞∑	∞∑	ADJ
ma-287	111	3	k=1	k=1	ADP
ma-287	111	4	1	1	NUM
ma-287	111	5	2k(1	2k(1	NOUN
ma-287	111	6	+	+	CCONJ
ma-287	111	7	x2	x2	PROPN
ma-287	111	8	−k	−k	PROPN
ma-287	111	9	)	)	PUNCT
ma-287	111	10	.	.	PUNCT
ma-287	112	1	as	as	SCONJ
ma-287	112	2	mentioned	mention	VERB
ma-287	112	3	in	in	ADP
ma-287	112	4	the	the	DET
ma-287	112	5	introduction	introduction	NOUN
ma-287	112	6	,	,	PUNCT
ma-287	112	7	this	this	DET
ma-287	112	8	result	result	NOUN
ma-287	112	9	is	be	AUX
ma-287	112	10	not	not	PART
ma-287	112	11	new	new	ADJ
ma-287	112	12	;	;	PUNCT
ma-287	112	13	it	it	PRON
ma-287	112	14	is	be	AUX
ma-287	112	15	a	a	DET
ma-287	112	16	famous	famous	ADJ
ma-287	112	17	result	result	NOUN
ma-287	112	18	proved	prove	VERB
ma-287	112	19	by	by	ADP
ma-287	112	20	srinivasa	srinivasa	PROPN
ma-287	112	21	ramanujan	ramanujan	PROPN
ma-287	112	22	,	,	PUNCT
ma-287	112	23	as	as	SCONJ
ma-287	112	24	highlighted	highlight	VERB
ma-287	112	25	in	in	ADP
ma-287	112	26	[	[	PUNCT
ma-287	112	27	2	2	NUM
ma-287	112	28	,	,	PUNCT
ma-287	112	29	chapter	chapter	NOUN
ma-287	112	30	31	31	NUM
ma-287	112	31	,	,	PUNCT
ma-287	112	32	page	page	NOUN
ma-287	112	33	399	399	NUM
ma-287	112	34	]	]	PUNCT
ma-287	112	35	and	and	CCONJ
ma-287	112	36	[	[	X
ma-287	112	37	5	5	NUM
ma-287	112	38	]	]	PUNCT
ma-287	112	39	.	.	PUNCT
ma-287	113	1	to	to	ADP
ma-287	113	2	the	the	DET
ma-287	113	3	best	good	ADJ
ma-287	113	4	of	of	ADP
ma-287	113	5	our	our	PRON
ma-287	113	6	knowledge	knowledge	NOUN
ma-287	113	7	,	,	PUNCT
ma-287	113	8	the	the	DET
ma-287	113	9	eight	eight	NUM
ma-287	113	10	results	result	NOUN
ma-287	113	11	below	below	ADV
ma-287	113	12	are	be	AUX
ma-287	113	13	new.(3	new.(3	NUM
ma-287	113	14	)	)	PUNCT
ma-287	113	15	for	for	ADP
ma-287	113	16	x	x	SYM
ma-287	113	17	>	>	X
ma-287	113	18	0	0	NUM
ma-287	113	19	,	,	PUNCT
ma-287	113	20	we	we	PRON
ma-287	113	21	have	have	VERB
ma-287	113	22	(	(	PUNCT
ma-287	114	1	x	x	SYM
ma-287	114	2	−	−	PROPN
ma-287	114	3	1)2	1)2	NUM
ma-287	115	1	−	−	PUNCT
ma-287	116	1	[	[	X
ma-287	116	2	log(x)]2	log(x)]2	X
ma-287	116	3	=	=	PUNCT
ma-287	117	1	+	+	ADP
ma-287	117	2	∞∑	∞∑	ADJ
ma-287	117	3	k=1	k=1	PROPN
ma-287	117	4	22(k−1)(x2	22(k−1)(x2	PROPN
ma-287	117	5	−k	−k	NOUN
ma-287	117	6	−	−	PROPN
ma-287	117	7	1)3(3	1)3(3	PROPN
ma-287	117	8	+	+	NUM
ma-287	117	9	x2−k	x2−k	PROPN
ma-287	117	10	)	)	PUNCT
ma-287	117	11	.	.	PUNCT
ma-287	118	1	(	(	PUNCT
ma-287	118	2	4	4	X
ma-287	118	3	)	)	PUNCT
ma-287	118	4	for	for	ADP
ma-287	118	5	x	x	X
ma-287	118	6	≥	≥	NOUN
ma-287	118	7	1	1	NUM
ma-287	118	8	,	,	PUNCT
ma-287	118	9	we	we	PRON
ma-287	118	10	have	have	VERB
ma-287	118	11	√	√	NUM
ma-287	118	12	x	x	PUNCT
ma-287	119	1	−	−	PROPN
ma-287	119	2	1−	1−	NUM
ma-287	119	3	√	√	NUM
ma-287	120	1	log(x	log(x	NUM
ma-287	120	2	)	)	PUNCT
ma-287	120	3	=	=	PUNCT
ma-287	121	1	+	+	PUNCT
ma-287	121	2	∞∑	∞∑	NUM
ma-287	121	3	k=1	k=1	PUNCT
ma-287	121	4	2(k−1)/2(x2	2(k−1)/2(x2	NOUN
ma-287	121	5	−k	−k	NOUN
ma-287	121	6	−	−	PROPN
ma-287	121	7	1)3/2√	1)3/2√	NUM
ma-287	121	8	1	1	NUM
ma-287	122	1	+	+	CCONJ
ma-287	122	2	x2	x2	NOUN
ma-287	122	3	−k	−k	PROPN
ma-287	122	4	+	+	CCONJ
ma-287	122	5	√	√	NUM
ma-287	122	6	2	2	NUM
ma-287	122	7	.	.	PUNCT
ma-287	123	1	(	(	PUNCT
ma-287	123	2	5	5	NUM
ma-287	123	3	)	)	PUNCT
ma-287	123	4	for	for	ADP
ma-287	123	5	x	x	SYM
ma-287	123	6	>	>	X
ma-287	123	7	1	1	NUM
ma-287	123	8	,	,	PUNCT
ma-287	123	9	we	we	PRON
ma-287	123	10	have	have	VERB
ma-287	124	1	1√	1√	NUM
ma-287	124	2	x	x	SYM
ma-287	124	3	−	−	PROPN
ma-287	124	4	1	1	NUM
ma-287	124	5	−	−	PROPN
ma-287	124	6	1√	1√	PROPN
ma-287	124	7	log(x	log(x	PROPN
ma-287	124	8	)	)	PUNCT
ma-287	124	9	=	=	PUNCT
ma-287	125	1	−	−	PUNCT
ma-287	126	1	+	+	NOUN
ma-287	126	2	∞∑	∞∑	ADJ
ma-287	126	3	k=1	k=1	ADP
ma-287	126	4	1	1	NUM
ma-287	126	5	2k/2	2k/2	NUM
ma-287	127	1	[	[	X
ma-287	127	2	√	√	ADJ
ma-287	127	3	1	1	NUM
ma-287	127	4	+	+	NUM
ma-287	127	5	x2	x2	NOUN
ma-287	127	6	−k	−k	PROPN
ma-287	127	7	+	+	CCONJ
ma-287	127	8	√	√	NUM
ma-287	127	9	2	2	NUM
ma-287	127	10	]	]	PUNCT
ma-287	127	11	√x2	√x2	NOUN
ma-287	127	12	−k	−k	PROPN
ma-287	127	13	−	−	NOUN
ma-287	127	14	1	1	NUM
ma-287	127	15	1	1	NUM
ma-287	127	16	+	+	NUM
ma-287	127	17	x2	x2	PROPN
ma-287	127	18	−k	−k	PROPN
ma-287	127	19	.	.	PUNCT
ma-287	128	1	(	(	PUNCT
ma-287	128	2	6	6	NUM
ma-287	128	3	)	)	PUNCT
ma-287	128	4	for	for	ADP
ma-287	128	5	x	x	SYM
ma-287	128	6	>	>	X
ma-287	128	7	1	1	NUM
ma-287	128	8	,	,	PUNCT
ma-287	128	9	we	we	PRON
ma-287	128	10	have	have	VERB
ma-287	128	11	log(x	log(x	NUM
ma-287	128	12	−	−	PROPN
ma-287	128	13	1)−	1)−	NUM
ma-287	128	14	log	log	NOUN
ma-287	129	1	[	[	X
ma-287	129	2	log(x	log(x	NOUN
ma-287	129	3	)	)	PUNCT
ma-287	129	4	]	]	PUNCT
ma-287	130	1	=	=	PUNCT
ma-287	131	1	+	+	PUNCT
ma-287	131	2	∞∑	∞∑	NOUN
ma-287	131	3	k=1	k=1	PUNCT
ma-287	131	4	log	log	NOUN
ma-287	131	5	(	(	PUNCT
ma-287	131	6	1	1	NUM
ma-287	131	7	+	+	NUM
ma-287	131	8	x2	x2	NOUN
ma-287	131	9	−k	−k	PROPN
ma-287	131	10	2	2	NUM
ma-287	131	11	)	)	PUNCT
ma-287	131	12	.	.	PUNCT
ma-287	132	1	(	(	PUNCT
ma-287	132	2	7	7	X
ma-287	132	3	)	)	PUNCT
ma-287	132	4	for	for	ADP
ma-287	132	5	x	x	SYM
ma-287	132	6	>	>	X
ma-287	132	7	0	0	NUM
ma-287	132	8	,	,	PUNCT
ma-287	132	9	we	we	PRON
ma-287	132	10	have	have	VERB
ma-287	132	11	sin(x	sin(x	PROPN
ma-287	132	12	−	−	PROPN
ma-287	132	13	1)−	1)−	PROPN
ma-287	132	14	sin[log(x	sin[log(x	NOUN
ma-287	132	15	)	)	PUNCT
ma-287	132	16	]	]	PUNCT
ma-287	133	1	=	=	PUNCT
ma-287	133	2	2	2	NUM
ma-287	133	3	+	+	ADP
ma-287	133	4	∞∑	∞∑	NUM
ma-287	133	5	k=1	k=1	PRON
ma-287	133	6	sin	sin	NOUN
ma-287	133	7	[	[	PUNCT
ma-287	133	8	2k−2(x2	2k−2(x2	NOUN
ma-287	133	9	−k	−k	NOUN
ma-287	133	10	−	−	PROPN
ma-287	133	11	1)2	1)2	NUM
ma-287	133	12	]	]	PUNCT
ma-287	134	1	cos	cos	PROPN
ma-287	134	2	[	[	PUNCT
ma-287	134	3	2k−2(x2	2k−2(x2	NOUN
ma-287	134	4	−k	−k	NOUN
ma-287	134	5	−	−	PROPN
ma-287	134	6	1)(3	1)(3	PROPN
ma-287	134	7	+	+	CCONJ
ma-287	134	8	x2−k	x2−k	PROPN
ma-287	134	9	)	)	PUNCT
ma-287	134	10	]	]	PUNCT
ma-287	134	11	.	.	PUNCT
ma-287	135	1	(	(	PUNCT
ma-287	135	2	8)	8)	NUM
ma-287	135	3	for	for	ADP
ma-287	135	4	x	x	SYM
ma-287	135	5	>	>	X
ma-287	135	6	0	0	NUM
ma-287	135	7	,	,	PUNCT
ma-287	135	8	we	we	PRON
ma-287	135	9	have	have	VERB
ma-287	135	10	cos(x	cos(x	PROPN
ma-287	135	11	−	−	PROPN
ma-287	135	12	1)−	1)−	NUM
ma-287	135	13	cos[log(x	cos[log(x	NOUN
ma-287	135	14	)	)	PUNCT
ma-287	135	15	]	]	PUNCT
ma-287	136	1	=	=	PUNCT
ma-287	136	2	−2	−2	NOUN
ma-287	137	1	+	+	NOUN
ma-287	137	2	∞∑	∞∑	ADJ
ma-287	137	3	k=1	k=1	PUNCT
ma-287	137	4	sin	sin	NOUN
ma-287	137	5	[	[	PUNCT
ma-287	137	6	2k−2(x2	2k−2(x2	NOUN
ma-287	137	7	−k	−k	NOUN
ma-287	137	8	−	−	PROPN
ma-287	137	9	1)2	1)2	NUM
ma-287	137	10	]	]	PUNCT
ma-287	137	11	sin	sin	NOUN
ma-287	137	12	[	[	PUNCT
ma-287	137	13	2k−2(x2	2k−2(x2	NOUN
ma-287	137	14	−k	−k	NOUN
ma-287	137	15	−	−	PROPN
ma-287	137	16	1)(3	1)(3	PROPN
ma-287	137	17	+	+	CCONJ
ma-287	137	18	x2−k	x2−k	PROPN
ma-287	137	19	)	)	PUNCT
ma-287	137	20	]	]	PUNCT
ma-287	137	21	.	.	PUNCT
ma-287	138	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	138	2	eur	eur	PROPN
ma-287	138	3	.	.	PUNCT
ma-287	139	1	j.	j.	PROPN
ma-287	139	2	math	math	PROPN
ma-287	139	3	.	.	PUNCT
ma-287	140	1	anal	anal	PROPN
ma-287	140	2	.	.	PUNCT
ma-287	141	1	10.28924	10.28924	NUM
ma-287	141	2	/	/	SYM
ma-287	141	3	ada	ada	PROPN
ma-287	141	4	/	/	SYM
ma-287	141	5	ma.5.12	ma.5.12	PROPN
ma-287	141	6	5(9	5(9	NUM
ma-287	141	7	)	)	PUNCT
ma-287	141	8	for	for	ADP
ma-287	141	9	x	x	SYM
ma-287	141	10	>	>	X
ma-287	141	11	0	0	NUM
ma-287	141	12	,	,	PUNCT
ma-287	141	13	we	we	PRON
ma-287	141	14	have	have	VERB
ma-287	141	15	sinh(x	sinh(x	NUM
ma-287	141	16	−	−	PROPN
ma-287	141	17	1)−	1)−	PROPN
ma-287	141	18	sinh[log(x	sinh[log(x	PROPN
ma-287	141	19	)	)	PUNCT
ma-287	141	20	]	]	PUNCT
ma-287	142	1	=	=	PUNCT
ma-287	142	2	2	2	NUM
ma-287	142	3	+	+	ADP
ma-287	142	4	∞∑	∞∑	NUM
ma-287	142	5	k=1	k=1	PROPN
ma-287	142	6	sinh	sinh	NOUN
ma-287	142	7	[	[	PUNCT
ma-287	142	8	2k−2(x2	2k−2(x2	NOUN
ma-287	142	9	−k	−k	NOUN
ma-287	142	10	−	−	PROPN
ma-287	142	11	1)2	1)2	NUM
ma-287	142	12	]	]	PUNCT
ma-287	142	13	cosh	cosh	PROPN
ma-287	142	14	[	[	PUNCT
ma-287	142	15	2k−2(x2	2k−2(x2	NOUN
ma-287	142	16	−k	−k	NOUN
ma-287	142	17	−	−	PROPN
ma-287	142	18	1)(3	1)(3	PROPN
ma-287	142	19	+	+	CCONJ
ma-287	142	20	x2−k	x2−k	PROPN
ma-287	142	21	)	)	PUNCT
ma-287	142	22	]	]	PUNCT
ma-287	142	23	.	.	PUNCT
ma-287	143	1	(	(	PUNCT
ma-287	143	2	10	10	NUM
ma-287	143	3	)	)	PUNCT
ma-287	143	4	for	for	ADP
ma-287	143	5	x	x	SYM
ma-287	143	6	>	>	X
ma-287	143	7	0	0	NUM
ma-287	143	8	,	,	PUNCT
ma-287	143	9	we	we	PRON
ma-287	143	10	have	have	VERB
ma-287	143	11	cosh(x	cosh(x	NUM
ma-287	143	12	−	−	PROPN
ma-287	143	13	1)−	1)−	PROPN
ma-287	143	14	cosh[log(x	cosh[log(x	NOUN
ma-287	143	15	)	)	PUNCT
ma-287	143	16	]	]	PUNCT
ma-287	144	1	=	=	PUNCT
ma-287	144	2	2	2	NUM
ma-287	144	3	+	+	ADP
ma-287	144	4	∞∑	∞∑	NUM
ma-287	144	5	k=1	k=1	PROPN
ma-287	144	6	sinh	sinh	NOUN
ma-287	144	7	[	[	PUNCT
ma-287	144	8	2k−2(x2	2k−2(x2	NOUN
ma-287	144	9	−k	−k	NOUN
ma-287	144	10	−	−	PROPN
ma-287	144	11	1)2	1)2	NUM
ma-287	144	12	]	]	PUNCT
ma-287	144	13	sinh	sinh	NOUN
ma-287	144	14	[	[	PUNCT
ma-287	144	15	2k−2(x2	2k−2(x2	NOUN
ma-287	144	16	−k	−k	NOUN
ma-287	144	17	−	−	PROPN
ma-287	144	18	1)(3	1)(3	PROPN
ma-287	144	19	+	+	CCONJ
ma-287	144	20	x2−k	x2−k	PROPN
ma-287	144	21	)	)	PUNCT
ma-287	144	22	]	]	PUNCT
ma-287	144	23	.	.	PUNCT
ma-287	145	1	proof	proof	NOUN
ma-287	145	2	.	.	PUNCT
ma-287	146	1	let	let	VERB
ma-287	146	2	us	we	PRON
ma-287	146	3	prove	prove	VERB
ma-287	146	4	each	each	DET
ma-287	146	5	result	result	NOUN
ma-287	146	6	,	,	PUNCT
ma-287	146	7	one	one	NUM
ma-287	146	8	by	by	ADP
ma-287	146	9	one.(1	one.(1	PROPN
ma-287	146	10	)	)	PUNCT
ma-287	146	11	for	for	ADP
ma-287	146	12	x	x	SYM
ma-287	146	13	>	>	X
ma-287	146	14	0	0	NUM
ma-287	146	15	,	,	PUNCT
ma-287	146	16	by	by	ADP
ma-287	146	17	applying	apply	VERB
ma-287	146	18	theorem	theorem	NOUN
ma-287	146	19	2.1	2.1	NUM
ma-287	146	20	with	with	ADP
ma-287	146	21	φ(t	φ(t	NOUN
ma-287	146	22	)	)	PUNCT
ma-287	147	1	=	=	SYM
ma-287	147	2	t	t	PROPN
ma-287	147	3	,	,	PUNCT
ma-287	147	4	we	we	PRON
ma-287	147	5	obtain	obtain	VERB
ma-287	147	6	x	x	PUNCT
ma-287	147	7	−	−	PROPN
ma-287	147	8	1−	1−	NUM
ma-287	147	9	log(x	log(x	PROPN
ma-287	147	10	)	)	PUNCT
ma-287	147	11	=	=	SYM
ma-287	148	1	φ(x	φ(x	PROPN
ma-287	148	2	−	−	PROPN
ma-287	148	3	1)−	1)−	NUM
ma-287	148	4	φ[log(x	φ[log(x	NUM
ma-287	148	5	)	)	PUNCT
ma-287	148	6	]	]	PUNCT
ma-287	149	1	=	=	PUNCT
ma-287	150	1	+	+	PUNCT
ma-287	150	2	∞∑	∞∑	NUM
ma-287	150	3	k=1	k=1	ADJ
ma-287	150	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	150	5	)	)	PUNCT
ma-287	150	6	,	,	PUNCT
ma-287	150	7	where	where	SCONJ
ma-287	150	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	150	9	)	)	PUNCT
ma-287	151	1	=	=	PUNCT
ma-287	151	2	φ	φ	PROPN
ma-287	151	3	[	[	PUNCT
ma-287	151	4	2k−1(x2	2k−1(x2	NOUN
ma-287	151	5	−(k−1	−(k−1	NOUN
ma-287	151	6	)	)	PUNCT
ma-287	152	1	−	−	PROPN
ma-287	152	2	1	1	NUM
ma-287	152	3	)	)	PUNCT
ma-287	152	4	]	]	PUNCT
ma-287	153	1	−	−	PROPN
ma-287	153	2	φ	φ	PROPN
ma-287	153	3	[	[	PUNCT
ma-287	153	4	2k(x2	2k(x2	NOUN
ma-287	153	5	−k	−k	ADJ
ma-287	153	6	−	−	NOUN
ma-287	153	7	1	1	NUM
ma-287	153	8	)	)	PUNCT
ma-287	153	9	]	]	PUNCT
ma-287	154	1	=	=	SYM
ma-287	154	2	2k−1(x2	2k−1(x2	NUM
ma-287	154	3	−(k−1	−(k−1	NOUN
ma-287	154	4	)	)	PUNCT
ma-287	155	1	−	−	PROPN
ma-287	156	1	1)−	1)−	NUM
ma-287	156	2	2k(x2−k	2k(x2−k	NUM
ma-287	156	3	−	−	NOUN
ma-287	156	4	1	1	NUM
ma-287	156	5	)	)	PUNCT
ma-287	156	6	=	=	SYM
ma-287	156	7	2k−1(x2	2k−1(x2	NOUN
ma-287	156	8	−k	−k	NOUN
ma-287	156	9	−	−	NOUN
ma-287	156	10	1)(1	1)(1	NUM
ma-287	156	11	+	+	CCONJ
ma-287	156	12	x2−k	x2−k	PROPN
ma-287	156	13	)	)	PUNCT
ma-287	157	1	−	−	PROPN
ma-287	158	1	2k(x2−k	2k(x2−k	NUM
ma-287	158	2	−	−	NOUN
ma-287	158	3	1	1	NUM
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ma-287	158	5	=	=	SYM
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ma-287	158	7	−k	−k	NOUN
ma-287	158	8	−	−	NOUN
ma-287	158	9	1	1	NUM
ma-287	158	10	)	)	PUNCT
ma-287	158	11	[	[	PUNCT
ma-287	158	12	(	(	PUNCT
ma-287	158	13	1	1	NUM
ma-287	158	14	+	+	NUM
ma-287	158	15	x2	x2	NOUN
ma-287	158	16	−k	−k	ADJ
ma-287	158	17	)	)	PUNCT
ma-287	159	1	−	−	PROPN
ma-287	159	2	2	2	NUM
ma-287	159	3	]	]	PUNCT
ma-287	159	4	=	=	SYM
ma-287	159	5	2k−1(x2	2k−1(x2	NOUN
ma-287	159	6	−k	−k	NOUN
ma-287	159	7	−	−	PROPN
ma-287	160	1	1)2	1)2	NUM
ma-287	160	2	.	.	PUNCT
ma-287	161	1	the	the	DET
ma-287	161	2	used	use	VERB
ma-287	161	3	factorization	factorization	NOUN
ma-287	161	4	arguments	argument	NOUN
ma-287	161	5	are	be	AUX
ma-287	161	6	the	the	DET
ma-287	161	7	same	same	ADJ
ma-287	161	8	as	as	ADP
ma-287	161	9	those	those	PRON
ma-287	161	10	in	in	ADP
ma-287	161	11	[	[	X
ma-287	161	12	4	4	NUM
ma-287	161	13	]	]	PUNCT
ma-287	161	14	.	.	PUNCT
ma-287	162	1	hence	hence	ADV
ma-287	162	2	,	,	PUNCT
ma-287	162	3	we	we	PRON
ma-287	162	4	have	have	VERB
ma-287	162	5	x	x	X
ma-287	162	6	−	−	PROPN
ma-287	162	7	1−	1−	NUM
ma-287	162	8	log(x	log(x	NUM
ma-287	162	9	)	)	PUNCT
ma-287	162	10	=	=	PUNCT
ma-287	163	1	+	+	PUNCT
ma-287	163	2	∞∑	∞∑	NUM
ma-287	163	3	k=1	k=1	ADJ
ma-287	163	4	2k−1(x2	2k−1(x2	NOUN
ma-287	163	5	−k	−k	NOUN
ma-287	163	6	−	−	PROPN
ma-287	163	7	1)2	1)2	NUM
ma-287	163	8	.	.	PUNCT
ma-287	164	1	(	(	PUNCT
ma-287	164	2	2	2	NUM
ma-287	164	3	)	)	PUNCT
ma-287	164	4	for	for	ADP
ma-287	164	5	x	x	SYM
ma-287	164	6	>	>	X
ma-287	164	7	0	0	NUM
ma-287	164	8	,	,	PUNCT
ma-287	164	9	applying	apply	VERB
ma-287	164	10	theorem	theorem	NOUN
ma-287	164	11	2.1	2.1	NUM
ma-287	164	12	to	to	ADP
ma-287	164	13	φ(t	φ(t	PROPN
ma-287	164	14	)	)	PUNCT
ma-287	164	15	=	=	SYM
ma-287	165	1	1	1	NUM
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ma-287	165	3	t	t	NOUN
ma-287	165	4	,	,	PUNCT
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ma-287	165	7	that	that	SCONJ
ma-287	165	8	1	1	NUM
ma-287	165	9	x	x	SYM
ma-287	165	10	−	−	NOUN
ma-287	165	11	1	1	NUM
ma-287	165	12	−	−	PROPN
ma-287	165	13	1	1	NUM
ma-287	165	14	log(x	log(x	NOUN
ma-287	165	15	)	)	PUNCT
ma-287	165	16	=	=	SYM
ma-287	166	1	φ(x	φ(x	PROPN
ma-287	166	2	−	−	PROPN
ma-287	166	3	1)−	1)−	NUM
ma-287	166	4	φ[log(x	φ[log(x	NUM
ma-287	166	5	)	)	PUNCT
ma-287	166	6	]	]	PUNCT
ma-287	167	1	=	=	PUNCT
ma-287	168	1	+	+	PUNCT
ma-287	168	2	∞∑	∞∑	NUM
ma-287	168	3	k=1	k=1	ADJ
ma-287	168	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	168	5	)	)	PUNCT
ma-287	168	6	,	,	PUNCT
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ma-287	169	1	=	=	PUNCT
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ma-287	169	4	2k−1(x2	2k−1(x2	NOUN
ma-287	169	5	−(k−1	−(k−1	NOUN
ma-287	169	6	)	)	PUNCT
ma-287	170	1	−	−	PROPN
ma-287	170	2	1	1	NUM
ma-287	170	3	)	)	PUNCT
ma-287	170	4	]	]	PUNCT
ma-287	171	1	−	−	PROPN
ma-287	171	2	φ	φ	PROPN
ma-287	171	3	[	[	PUNCT
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ma-287	171	6	−	−	NOUN
ma-287	171	7	1	1	NUM
ma-287	171	8	)	)	PUNCT
ma-287	171	9	]	]	PUNCT
ma-287	172	1	=	=	PUNCT
ma-287	172	2	1	1	NUM
ma-287	172	3	2k−1(x2−(k−1	2k−1(x2−(k−1	NUM
ma-287	172	4	)	)	PUNCT
ma-287	172	5	−	−	NUM
ma-287	172	6	1	1	NUM
ma-287	172	7	)	)	PUNCT
ma-287	172	8	−	−	NOUN
ma-287	172	9	1	1	NUM
ma-287	172	10	2k(x2	2k(x2	NOUN
ma-287	172	11	−k	−k	ADJ
ma-287	172	12	−	−	NOUN
ma-287	172	13	1	1	NUM
ma-287	172	14	)	)	PUNCT
ma-287	172	15	=	=	SYM
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ma-287	172	18	−(k−1	−(k−1	NOUN
ma-287	172	19	)	)	PUNCT
ma-287	173	1	−	−	PROPN
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ma-287	174	4	1	1	NUM
ma-287	174	5	)	)	PUNCT
ma-287	174	6	22k−1(x2−(k−1	22k−1(x2−(k−1	NUM
ma-287	174	7	)	)	PUNCT
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ma-287	175	1	1)(x2−k	1)(x2−k	NUM
ma-287	176	1	−	−	NOUN
ma-287	176	2	1	1	NUM
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ma-287	176	4	=	=	SYM
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ma-287	178	1	−	−	NOUN
ma-287	178	2	1	1	NUM
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ma-287	178	4	=	=	SYM
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ma-287	179	2	(	(	PUNCT
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ma-287	179	5	−	−	PROPN
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ma-287	179	7	2k(1	2k(1	NOUN
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ma-287	179	9	x2	x2	PROPN
ma-287	179	10	−k	−k	PROPN
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ma-287	180	1	(	(	PUNCT
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ma-287	181	2	=	=	SYM
ma-287	181	3	−	−	PROPN
ma-287	181	4	1	1	NUM
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ma-287	181	6	+	+	CCONJ
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ma-287	184	2	.	.	PUNCT
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ma-287	185	3	ada	ada	PROPN
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ma-287	186	1	6hence	6hence	NUM
ma-287	186	2	,	,	PUNCT
ma-287	186	3	we	we	PRON
ma-287	186	4	have	have	VERB
ma-287	186	5	1	1	NUM
ma-287	186	6	x	x	SYM
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ma-287	186	8	1	1	NUM
ma-287	186	9	−	−	PROPN
ma-287	186	10	1	1	NUM
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ma-287	187	1	−	−	PUNCT
ma-287	188	1	+	+	NOUN
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ma-287	189	13	φ(t	φ(t	PROPN
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ma-287	189	20	(	(	PUNCT
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ma-287	190	2	−	−	PUNCT
ma-287	191	1	[	[	X
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ma-287	191	4	φ(x	φ(x	PROPN
ma-287	191	5	−	−	PROPN
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ma-287	192	1	=	=	PUNCT
ma-287	193	1	+	+	PUNCT
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ma-287	193	4	αk(φ)(x	αk(φ)(x	PROPN
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ma-287	193	6	,	,	PUNCT
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ma-287	194	5	−(k−1	−(k−1	NOUN
ma-287	194	6	)	)	PUNCT
ma-287	195	1	−	−	PROPN
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ma-287	196	1	−	−	PROPN
ma-287	196	2	φ	φ	PROPN
ma-287	196	3	[	[	PUNCT
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ma-287	196	7	1	1	NUM
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ma-287	197	1	=	=	PUNCT
ma-287	197	2	[	[	PUNCT
ma-287	197	3	2k−1(x2	2k−1(x2	NUM
ma-287	197	4	−(k−1	−(k−1	NOUN
ma-287	197	5	)	)	PUNCT
ma-287	197	6	−	−	PROPN
ma-287	198	1	1	1	NUM
ma-287	198	2	)	)	PUNCT
ma-287	198	3	]	]	PUNCT
ma-287	198	4	2	2	NUM
ma-287	198	5	−	−	NOUN
ma-287	198	6	[	[	PUNCT
ma-287	198	7	2k(x2	2k(x2	NOUN
ma-287	198	8	−k	−k	ADJ
ma-287	198	9	−	−	NOUN
ma-287	198	10	1	1	NUM
ma-287	198	11	)	)	PUNCT
ma-287	198	12	]	]	PUNCT
ma-287	198	13	2	2	X
ma-287	198	14	=	=	SYM
ma-287	198	15	[	[	PUNCT
ma-287	198	16	2k−1(x2	2k−1(x2	NOUN
ma-287	198	17	−(k−1	−(k−1	NOUN
ma-287	198	18	)	)	PUNCT
ma-287	198	19	−	−	PROPN
ma-287	199	1	1)−	1)−	NUM
ma-287	199	2	2k(x2−k	2k(x2−k	NUM
ma-287	199	3	−	−	NOUN
ma-287	199	4	1	1	NUM
ma-287	199	5	)	)	PUNCT
ma-287	199	6	]	]	PUNCT
ma-287	200	1	[	[	PUNCT
ma-287	200	2	2k−1(x2	2k−1(x2	NOUN
ma-287	200	3	−(k−1	−(k−1	NOUN
ma-287	200	4	)	)	PUNCT
ma-287	201	1	−	−	NOUN
ma-287	201	2	1	1	NUM
ma-287	201	3	)	)	PUNCT
ma-287	202	1	+	+	CCONJ
ma-287	202	2	2k(x2−k	2k(x2−k	NUM
ma-287	202	3	−	−	NOUN
ma-287	202	4	1	1	NUM
ma-287	202	5	)	)	PUNCT
ma-287	202	6	]	]	PUNCT
ma-287	203	1	=	=	SYM
ma-287	203	2	2k−1(x2	2k−1(x2	NOUN
ma-287	203	3	−k	−k	NOUN
ma-287	203	4	−	−	PROPN
ma-287	203	5	1)22k−1(x2−k	1)22k−1(x2−k	NUM
ma-287	203	6	−	−	PROPN
ma-287	203	7	1)(3	1)(3	NUM
ma-287	203	8	+	+	CCONJ
ma-287	203	9	x2−k	x2−k	PROPN
ma-287	203	10	)	)	PUNCT
ma-287	204	1	=	=	SYM
ma-287	204	2	22(k−1)(x2	22(k−1)(x2	NOUN
ma-287	204	3	−k	−k	NOUN
ma-287	204	4	−	−	PROPN
ma-287	205	1	1)3(3	1)3(3	PROPN
ma-287	205	2	+	+	NUM
ma-287	205	3	x2−k	x2−k	PROPN
ma-287	205	4	)	)	PUNCT
ma-287	205	5	.	.	PUNCT
ma-287	206	1	hence	hence	ADV
ma-287	206	2	,	,	PUNCT
ma-287	206	3	we	we	PRON
ma-287	206	4	have	have	VERB
ma-287	206	5	(	(	PUNCT
ma-287	206	6	x	x	SYM
ma-287	206	7	−	−	PROPN
ma-287	206	8	1)2	1)2	NUM
ma-287	206	9	−	−	PUNCT
ma-287	207	1	[	[	X
ma-287	207	2	log(x)]2	log(x)]2	X
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ma-287	208	6	−	−	PROPN
ma-287	208	7	1)3(3	1)3(3	PROPN
ma-287	208	8	+	+	NUM
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ma-287	209	1	(	(	PUNCT
ma-287	209	2	4	4	X
ma-287	209	3	)	)	PUNCT
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ma-287	209	5	x	x	X
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ma-287	209	7	1	1	NUM
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ma-287	209	13	φ(t	φ(t	NOUN
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ma-287	209	15	=	=	SYM
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ma-287	210	2	,	,	PUNCT
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ma-287	210	5	√	√	NUM
ma-287	210	6	x	x	PUNCT
ma-287	210	7	−	−	PROPN
ma-287	210	8	1−	1−	NUM
ma-287	210	9	√	√	NUM
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ma-287	210	11	)	)	PUNCT
ma-287	210	12	=	=	SYM
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ma-287	211	2	−	−	PROPN
ma-287	211	3	1)−	1)−	NUM
ma-287	211	4	φ[log(x	φ[log(x	NUM
ma-287	211	5	)	)	PUNCT
ma-287	211	6	]	]	PUNCT
ma-287	212	1	=	=	PUNCT
ma-287	213	1	+	+	PUNCT
ma-287	213	2	∞∑	∞∑	NUM
ma-287	213	3	k=1	k=1	ADJ
ma-287	213	4	αk(φ)(x	αk(φ)(x	PROPN
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ma-287	214	3	[	[	PUNCT
ma-287	214	4	2k−1(x2	2k−1(x2	NOUN
ma-287	214	5	−(k−1	−(k−1	NOUN
ma-287	214	6	)	)	PUNCT
ma-287	215	1	−	−	PROPN
ma-287	215	2	1	1	NUM
ma-287	215	3	)	)	PUNCT
ma-287	215	4	]	]	PUNCT
ma-287	216	1	−	−	PROPN
ma-287	216	2	φ	φ	PROPN
ma-287	216	3	[	[	PUNCT
ma-287	216	4	2k(x2	2k(x2	NOUN
ma-287	216	5	−k	−k	ADJ
ma-287	216	6	−	−	NOUN
ma-287	216	7	1	1	NUM
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ma-287	216	9	]	]	PUNCT
ma-287	217	1	=	=	PUNCT
ma-287	218	1	√	√	NUM
ma-287	218	2	2k−1(x2−(k−1	2k−1(x2−(k−1	NUM
ma-287	218	3	)	)	PUNCT
ma-287	219	1	−	−	PROPN
ma-287	219	2	1)−	1)−	NUM
ma-287	219	3	√	√	PROPN
ma-287	219	4	2k(x2	2k(x2	NOUN
ma-287	219	5	−k	−k	NOUN
ma-287	219	6	−	−	NOUN
ma-287	219	7	1	1	NUM
ma-287	219	8	)	)	PUNCT
ma-287	219	9	=	=	SYM
ma-287	219	10	2k−1(x2	2k−1(x2	NOUN
ma-287	219	11	−(k−1	−(k−1	NOUN
ma-287	219	12	)	)	PUNCT
ma-287	220	1	−	−	PROPN
ma-287	221	1	1)−	1)−	NUM
ma-287	221	2	2k(x2−k	2k(x2−k	NUM
ma-287	221	3	−	−	NUM
ma-287	221	4	1)√	1)√	NUM
ma-287	221	5	2k−1(x2−(k−1	2k−1(x2−(k−1	NUM
ma-287	221	6	)	)	PUNCT
ma-287	221	7	−	−	NUM
ma-287	221	8	1	1	NUM
ma-287	221	9	)	)	PUNCT
ma-287	222	1	+	+	CCONJ
ma-287	222	2	√	√	NUM
ma-287	222	3	2k(x2	2k(x2	NOUN
ma-287	222	4	−k	−k	NOUN
ma-287	222	5	−	−	NOUN
ma-287	222	6	1	1	NUM
ma-287	222	7	)	)	PUNCT
ma-287	222	8	=	=	SYM
ma-287	222	9	2k−1(x2	2k−1(x2	NOUN
ma-287	222	10	−k	−k	NOUN
ma-287	222	11	−	−	X
ma-287	222	12	1)2√	1)2√	NUM
ma-287	222	13	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	222	14	−	−	PROPN
ma-287	222	15	1)(1	1)(1	NUM
ma-287	222	16	+	+	CCONJ
ma-287	222	17	x2−k	x2−k	PROPN
ma-287	222	18	)	)	PUNCT
ma-287	223	1	+	+	CCONJ
ma-287	223	2	√	√	NUM
ma-287	223	3	2[2k−1(x2−k	2[2k−1(x2−k	NUM
ma-287	223	4	−	−	NOUN
ma-287	223	5	1	1	NUM
ma-287	223	6	)	)	PUNCT
ma-287	223	7	]	]	PUNCT
ma-287	224	1	=	=	PUNCT
ma-287	224	2	2(k−1)/2(x2	2(k−1)/2(x2	NUM
ma-287	224	3	−k	−k	NOUN
ma-287	224	4	−	−	PROPN
ma-287	224	5	1)3/2√	1)3/2√	NUM
ma-287	224	6	1	1	NUM
ma-287	224	7	+	+	CCONJ
ma-287	224	8	x2	x2	NOUN
ma-287	224	9	−k	−k	PROPN
ma-287	224	10	+	+	CCONJ
ma-287	224	11	√	√	NUM
ma-287	224	12	2	2	NUM
ma-287	224	13	.	.	PUNCT
ma-287	225	1	hence	hence	ADV
ma-287	225	2	,	,	PUNCT
ma-287	225	3	we	we	PRON
ma-287	225	4	have	have	VERB
ma-287	225	5	√	√	NUM
ma-287	225	6	x	x	PUNCT
ma-287	225	7	−	−	PROPN
ma-287	225	8	1−	1−	NUM
ma-287	225	9	√	√	NUM
ma-287	225	10	log(x	log(x	NUM
ma-287	225	11	)	)	PUNCT
ma-287	225	12	=	=	PUNCT
ma-287	226	1	+	+	PUNCT
ma-287	226	2	∞∑	∞∑	NUM
ma-287	226	3	k=1	k=1	PUNCT
ma-287	226	4	2(k−1)/2(x2	2(k−1)/2(x2	NOUN
ma-287	226	5	−k	−k	NOUN
ma-287	226	6	−	−	PROPN
ma-287	226	7	1)3/2√	1)3/2√	NUM
ma-287	226	8	1	1	NUM
ma-287	227	1	+	+	CCONJ
ma-287	227	2	x2	x2	NOUN
ma-287	227	3	−k	−k	PROPN
ma-287	227	4	+	+	CCONJ
ma-287	227	5	√	√	NUM
ma-287	227	6	2	2	NUM
ma-287	227	7	.	.	PUNCT
ma-287	228	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	228	2	eur	eur	PROPN
ma-287	228	3	.	.	PUNCT
ma-287	229	1	j.	j.	PROPN
ma-287	229	2	math	math	PROPN
ma-287	229	3	.	.	PUNCT
ma-287	230	1	anal	anal	PROPN
ma-287	230	2	.	.	PUNCT
ma-287	231	1	10.28924	10.28924	NUM
ma-287	231	2	/	/	SYM
ma-287	231	3	ada	ada	PROPN
ma-287	231	4	/	/	SYM
ma-287	231	5	ma.5.12	ma.5.12	PROPN
ma-287	231	6	7(5	7(5	NUM
ma-287	231	7	)	)	PUNCT
ma-287	231	8	for	for	ADP
ma-287	231	9	x	x	SYM
ma-287	231	10	>	>	X
ma-287	231	11	1	1	NUM
ma-287	231	12	,	,	PUNCT
ma-287	231	13	applying	apply	VERB
ma-287	231	14	theorem	theorem	NOUN
ma-287	231	15	2.1	2.1	NUM
ma-287	231	16	to	to	ADP
ma-287	231	17	φ(t	φ(t	PROPN
ma-287	231	18	)	)	PUNCT
ma-287	231	19	=	=	SYM
ma-287	231	20	1/√t	1/√t	NUM
ma-287	231	21	,	,	PUNCT
ma-287	231	22	we	we	PRON
ma-287	231	23	have	have	VERB
ma-287	232	1	1√	1√	NUM
ma-287	232	2	x	x	SYM
ma-287	232	3	−	−	PROPN
ma-287	232	4	1	1	NUM
ma-287	232	5	−	−	PROPN
ma-287	232	6	1√	1√	PROPN
ma-287	232	7	log(x	log(x	PROPN
ma-287	232	8	)	)	PUNCT
ma-287	232	9	=	=	SYM
ma-287	232	10	φ(x	φ(x	PROPN
ma-287	232	11	−	−	PROPN
ma-287	232	12	1)−	1)−	NUM
ma-287	232	13	φ[log(x	φ[log(x	NUM
ma-287	232	14	)	)	PUNCT
ma-287	232	15	]	]	PUNCT
ma-287	233	1	=	=	PUNCT
ma-287	234	1	+	+	PUNCT
ma-287	234	2	∞∑	∞∑	NUM
ma-287	234	3	k=1	k=1	ADJ
ma-287	234	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	234	5	)	)	PUNCT
ma-287	234	6	,	,	PUNCT
ma-287	234	7	where	where	SCONJ
ma-287	234	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	234	9	)	)	PUNCT
ma-287	235	1	=	=	PUNCT
ma-287	235	2	φ	φ	PROPN
ma-287	235	3	[	[	PUNCT
ma-287	235	4	2k−1(x2	2k−1(x2	NOUN
ma-287	235	5	−(k−1	−(k−1	NOUN
ma-287	235	6	)	)	PUNCT
ma-287	236	1	−	−	PROPN
ma-287	236	2	1	1	NUM
ma-287	236	3	)	)	PUNCT
ma-287	236	4	]	]	PUNCT
ma-287	237	1	−	−	PROPN
ma-287	237	2	φ	φ	PROPN
ma-287	237	3	[	[	PUNCT
ma-287	237	4	2k(x2	2k(x2	NOUN
ma-287	237	5	−k	−k	ADJ
ma-287	237	6	−	−	NOUN
ma-287	237	7	1	1	NUM
ma-287	237	8	)	)	PUNCT
ma-287	237	9	]	]	PUNCT
ma-287	238	1	=	=	PUNCT
ma-287	238	2	1√	1√	NUM
ma-287	238	3	2k−1(x2−(k−1	2k−1(x2−(k−1	NUM
ma-287	238	4	)	)	PUNCT
ma-287	238	5	−	−	NUM
ma-287	238	6	1	1	NUM
ma-287	238	7	)	)	PUNCT
ma-287	238	8	−	−	PROPN
ma-287	238	9	1√	1√	PROPN
ma-287	238	10	2k(x2	2k(x2	NUM
ma-287	238	11	−k	−k	NOUN
ma-287	238	12	−	−	NOUN
ma-287	238	13	1	1	NUM
ma-287	238	14	)	)	PUNCT
ma-287	238	15	=	=	SYM
ma-287	239	1	−	−	PROPN
ma-287	239	2	√	√	NUM
ma-287	239	3	2k−1(x2−(k−1	2k−1(x2−(k−1	NUM
ma-287	239	4	)	)	PUNCT
ma-287	239	5	−	−	PROPN
ma-287	240	1	1)−	1)−	NUM
ma-287	240	2	√	√	PROPN
ma-287	240	3	2k(x2	2k(x2	NOUN
ma-287	240	4	−k	−k	PROPN
ma-287	240	5	−	−	PROPN
ma-287	240	6	1)√	1)√	NUM
ma-287	240	7	22k−1(x2−(k−1	22k−1(x2−(k−1	NUM
ma-287	240	8	)	)	PUNCT
ma-287	240	9	−	−	NOUN
ma-287	240	10	1)(x2−k	1)(x2−k	NUM
ma-287	241	1	−	−	NOUN
ma-287	241	2	1	1	NUM
ma-287	241	3	)	)	PUNCT
ma-287	241	4	=	=	SYM
ma-287	242	1	−	−	PROPN
ma-287	242	2	2k−1(x2	2k−1(x2	NOUN
ma-287	242	3	−(k−1	−(k−1	NOUN
ma-287	242	4	)	)	PUNCT
ma-287	243	1	−	−	PROPN
ma-287	244	1	1)−	1)−	NUM
ma-287	244	2	2k(x2−k	2k(x2−k	NUM
ma-287	244	3	−	−	NOUN
ma-287	244	4	1)[√	1)[√	NUM
ma-287	244	5	2k−1(x2−(k−1	2k−1(x2−(k−1	NUM
ma-287	244	6	)	)	PUNCT
ma-287	244	7	−	−	NUM
ma-287	244	8	1	1	NUM
ma-287	244	9	)	)	PUNCT
ma-287	245	1	+	+	CCONJ
ma-287	245	2	√	√	NUM
ma-287	245	3	2k(x2	2k(x2	NOUN
ma-287	245	4	−k	−k	NOUN
ma-287	245	5	−	−	NOUN
ma-287	245	6	1	1	NUM
ma-287	245	7	)	)	PUNCT
ma-287	245	8	]	]	X
ma-287	245	9	√	√	NUM
ma-287	245	10	22k−1(x2−(k−1	22k−1(x2−(k−1	NUM
ma-287	245	11	)	)	PUNCT
ma-287	245	12	−	−	PROPN
ma-287	245	13	1)(x2−k	1)(x2−k	NUM
ma-287	246	1	−	−	NOUN
ma-287	246	2	1	1	NUM
ma-287	246	3	)	)	PUNCT
ma-287	246	4	=	=	SYM
ma-287	247	1	−	−	PROPN
ma-287	247	2	2k−1(x2	2k−1(x2	NOUN
ma-287	247	3	−k	−k	NOUN
ma-287	247	4	−	−	PROPN
ma-287	247	5	1)2[√	1)2[√	NUM
ma-287	247	6	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	247	7	−	−	PROPN
ma-287	247	8	1)(1	1)(1	NUM
ma-287	247	9	+	+	CCONJ
ma-287	247	10	x2−k	x2−k	PROPN
ma-287	247	11	)	)	PUNCT
ma-287	248	1	+	+	CCONJ
ma-287	248	2	√	√	NUM
ma-287	248	3	2[2k−1(x2−k	2[2k−1(x2−k	NUM
ma-287	248	4	−	−	NOUN
ma-287	248	5	1	1	NUM
ma-287	248	6	)	)	PUNCT
ma-287	248	7	]	]	PUNCT
ma-287	249	1	]	]	PUNCT
ma-287	249	2	√	√	NUM
ma-287	249	3	22k−1(x2−k	22k−1(x2−k	NUM
ma-287	249	4	−	−	PROPN
ma-287	249	5	1)2(1	1)2(1	NUM
ma-287	249	6	+	+	CCONJ
ma-287	249	7	x2−k	x2−k	PROPN
ma-287	249	8	)	)	PUNCT
ma-287	250	1	=	=	PUNCT
ma-287	251	1	−	−	PROPN
ma-287	251	2	1	1	NUM
ma-287	251	3	2k/2	2k/2	NUM
ma-287	252	1	[	[	X
ma-287	252	2	√	√	ADJ
ma-287	252	3	1	1	NUM
ma-287	252	4	+	+	NUM
ma-287	252	5	x2	x2	NOUN
ma-287	252	6	−k	−k	PROPN
ma-287	252	7	+	+	CCONJ
ma-287	252	8	√	√	NUM
ma-287	252	9	2	2	NUM
ma-287	252	10	]	]	PUNCT
ma-287	252	11	√x2	√x2	NOUN
ma-287	252	12	−k	−k	PROPN
ma-287	252	13	−	−	NOUN
ma-287	252	14	1	1	NUM
ma-287	252	15	1	1	NUM
ma-287	252	16	+	+	NUM
ma-287	252	17	x2	x2	PROPN
ma-287	252	18	−k	−k	PROPN
ma-287	252	19	.	.	PUNCT
ma-287	253	1	hence	hence	ADV
ma-287	253	2	,	,	PUNCT
ma-287	253	3	we	we	PRON
ma-287	253	4	have	have	VERB
ma-287	253	5	1√	1√	NUM
ma-287	253	6	x	x	SYM
ma-287	253	7	−	−	PROPN
ma-287	253	8	1	1	NUM
ma-287	253	9	−	−	PROPN
ma-287	253	10	1√	1√	PROPN
ma-287	253	11	log(x	log(x	PROPN
ma-287	253	12	)	)	PUNCT
ma-287	253	13	=	=	PUNCT
ma-287	254	1	−	−	PUNCT
ma-287	255	1	+	+	NOUN
ma-287	255	2	∞∑	∞∑	ADJ
ma-287	255	3	k=1	k=1	ADP
ma-287	255	4	1	1	NUM
ma-287	255	5	2k/2	2k/2	NUM
ma-287	256	1	[	[	X
ma-287	256	2	√	√	ADJ
ma-287	256	3	1	1	NUM
ma-287	256	4	+	+	NUM
ma-287	256	5	x2	x2	NOUN
ma-287	256	6	−k	−k	PROPN
ma-287	256	7	+	+	CCONJ
ma-287	256	8	√	√	NUM
ma-287	256	9	2	2	NUM
ma-287	256	10	]	]	PUNCT
ma-287	256	11	√x2	√x2	NOUN
ma-287	256	12	−k	−k	PROPN
ma-287	256	13	−	−	NOUN
ma-287	256	14	1	1	NUM
ma-287	256	15	1	1	NUM
ma-287	256	16	+	+	NUM
ma-287	256	17	x2	x2	PROPN
ma-287	256	18	−k	−k	PROPN
ma-287	256	19	.	.	PUNCT
ma-287	257	1	(	(	PUNCT
ma-287	257	2	6	6	NUM
ma-287	257	3	)	)	PUNCT
ma-287	257	4	for	for	ADP
ma-287	257	5	x	x	SYM
ma-287	257	6	>	>	X
ma-287	257	7	1	1	NUM
ma-287	257	8	,	,	PUNCT
ma-287	257	9	it	it	PRON
ma-287	257	10	follows	follow	VERB
ma-287	257	11	from	from	ADP
ma-287	257	12	theorem	theorem	ADJ
ma-287	257	13	2.1	2.1	NUM
ma-287	257	14	with	with	ADP
ma-287	257	15	φ(t	φ(t	NOUN
ma-287	257	16	)	)	PUNCT
ma-287	258	1	=	=	SYM
ma-287	258	2	log(t	log(t	PROPN
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ma-287	259	2	log(x	log(x	PROPN
ma-287	260	1	−	−	PROPN
ma-287	260	2	1)−	1)−	NUM
ma-287	260	3	log[log(x	log[log(x	NOUN
ma-287	260	4	)	)	PUNCT
ma-287	260	5	]	]	PUNCT
ma-287	261	1	=	=	PUNCT
ma-287	261	2	φ(x	φ(x	PROPN
ma-287	261	3	−	−	PROPN
ma-287	261	4	1)−	1)−	NUM
ma-287	261	5	φ[log(x	φ[log(x	NUM
ma-287	261	6	)	)	PUNCT
ma-287	261	7	]	]	PUNCT
ma-287	262	1	=	=	PUNCT
ma-287	263	1	+	+	PUNCT
ma-287	263	2	∞∑	∞∑	NUM
ma-287	263	3	k=1	k=1	ADJ
ma-287	263	4	αk(φ)(x	αk(φ)(x	PROPN
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ma-287	263	6	,	,	PUNCT
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ma-287	263	8	αk(φ)(x	αk(φ)(x	NOUN
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ma-287	264	1	=	=	PUNCT
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ma-287	264	3	[	[	PUNCT
ma-287	264	4	2k−1(x2	2k−1(x2	NOUN
ma-287	264	5	−(k−1	−(k−1	NOUN
ma-287	264	6	)	)	PUNCT
ma-287	265	1	−	−	PROPN
ma-287	265	2	1	1	NUM
ma-287	265	3	)	)	PUNCT
ma-287	265	4	]	]	PUNCT
ma-287	266	1	−	−	PROPN
ma-287	266	2	φ	φ	PROPN
ma-287	266	3	[	[	PUNCT
ma-287	266	4	2k(x2	2k(x2	NOUN
ma-287	266	5	−k	−k	ADJ
ma-287	266	6	−	−	NOUN
ma-287	266	7	1	1	NUM
ma-287	266	8	)	)	PUNCT
ma-287	266	9	]	]	PUNCT
ma-287	267	1	=	=	PUNCT
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ma-287	267	3	[	[	PUNCT
ma-287	267	4	2k−1(x2	2k−1(x2	NOUN
ma-287	267	5	−(k−1	−(k−1	NOUN
ma-287	267	6	)	)	PUNCT
ma-287	267	7	−	−	PROPN
ma-287	267	8	1	1	NUM
ma-287	267	9	)	)	PUNCT
ma-287	267	10	]	]	PUNCT
ma-287	268	1	−	−	PROPN
ma-287	268	2	log	log	NOUN
ma-287	268	3	[	[	PUNCT
ma-287	268	4	2k(x2	2k(x2	NOUN
ma-287	268	5	−k	−k	ADJ
ma-287	268	6	−	−	NOUN
ma-287	268	7	1	1	NUM
ma-287	268	8	)	)	PUNCT
ma-287	268	9	]	]	PUNCT
ma-287	269	1	=	=	PUNCT
ma-287	269	2	log	log	NOUN
ma-287	269	3	[	[	PUNCT
ma-287	269	4	2k−1(x2	2k−1(x2	NOUN
ma-287	269	5	−(k−1	−(k−1	NOUN
ma-287	269	6	)	)	PUNCT
ma-287	269	7	−	−	NOUN
ma-287	269	8	1	1	NUM
ma-287	269	9	)	)	PUNCT
ma-287	269	10	2k(x2	2k(x2	NOUN
ma-287	269	11	−k	−k	ADJ
ma-287	269	12	−	−	NOUN
ma-287	269	13	1	1	NUM
ma-287	269	14	)	)	PUNCT
ma-287	269	15	]	]	PUNCT
ma-287	270	1	=	=	PUNCT
ma-287	270	2	log	log	NOUN
ma-287	270	3	[	[	PUNCT
ma-287	270	4	(	(	PUNCT
ma-287	270	5	x2	x2	NOUN
ma-287	270	6	−k	−k	VERB
ma-287	270	7	−	−	NOUN
ma-287	270	8	1)(1	1)(1	NUM
ma-287	270	9	+	+	CCONJ
ma-287	270	10	x2−k	x2−k	PROPN
ma-287	270	11	)	)	PUNCT
ma-287	270	12	2(x2	2(x2	NUM
ma-287	270	13	−k	−k	ADJ
ma-287	270	14	−	−	NOUN
ma-287	270	15	1	1	NUM
ma-287	270	16	)	)	PUNCT
ma-287	270	17	]	]	PUNCT
ma-287	271	1	=	=	PUNCT
ma-287	271	2	log	log	NOUN
ma-287	271	3	(	(	PUNCT
ma-287	271	4	1	1	NUM
ma-287	271	5	+	+	NUM
ma-287	271	6	x2	x2	NOUN
ma-287	271	7	−k	−k	PROPN
ma-287	271	8	2	2	NUM
ma-287	271	9	)	)	PUNCT
ma-287	271	10	.	.	PUNCT
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ma-287	272	2	eur	eur	PROPN
ma-287	272	3	.	.	PUNCT
ma-287	273	1	j.	j.	PROPN
ma-287	273	2	math	math	PROPN
ma-287	273	3	.	.	PUNCT
ma-287	274	1	anal	anal	PROPN
ma-287	274	2	.	.	PUNCT
ma-287	275	1	10.28924	10.28924	NUM
ma-287	275	2	/	/	SYM
ma-287	275	3	ada	ada	PROPN
ma-287	275	4	/	/	SYM
ma-287	275	5	ma.5.12	ma.5.12	PROPN
ma-287	275	6	8hence	8hence	NUM
ma-287	275	7	,	,	PUNCT
ma-287	275	8	we	we	PRON
ma-287	275	9	have	have	VERB
ma-287	275	10	log(x	log(x	NUM
ma-287	275	11	−	−	PROPN
ma-287	275	12	1)−	1)−	NUM
ma-287	275	13	log	log	NOUN
ma-287	276	1	[	[	X
ma-287	276	2	log(x	log(x	NOUN
ma-287	276	3	)	)	PUNCT
ma-287	276	4	]	]	PUNCT
ma-287	277	1	=	=	PUNCT
ma-287	278	1	+	+	PUNCT
ma-287	278	2	∞∑	∞∑	NOUN
ma-287	278	3	k=1	k=1	PUNCT
ma-287	278	4	log	log	NOUN
ma-287	278	5	(	(	PUNCT
ma-287	278	6	1	1	NUM
ma-287	278	7	+	+	NUM
ma-287	278	8	x2	x2	NOUN
ma-287	278	9	−k	−k	PROPN
ma-287	278	10	2	2	NUM
ma-287	278	11	)	)	PUNCT
ma-287	278	12	.	.	PUNCT
ma-287	279	1	(	(	PUNCT
ma-287	279	2	7	7	X
ma-287	279	3	)	)	PUNCT
ma-287	279	4	for	for	ADP
ma-287	279	5	x	x	SYM
ma-287	279	6	>	>	X
ma-287	279	7	0	0	NUM
ma-287	279	8	,	,	PUNCT
ma-287	279	9	applying	apply	VERB
ma-287	279	10	theorem	theorem	NOUN
ma-287	279	11	2.1	2.1	NUM
ma-287	279	12	to	to	ADP
ma-287	279	13	φ(t	φ(t	PROPN
ma-287	279	14	)	)	PUNCT
ma-287	279	15	=	=	SYM
ma-287	279	16	sin(t	sin(t	PROPN
ma-287	279	17	)	)	PUNCT
ma-287	279	18	,	,	PUNCT
ma-287	279	19	we	we	PRON
ma-287	279	20	obtain	obtain	VERB
ma-287	279	21	sin(x	sin(x	PROPN
ma-287	279	22	−	−	PROPN
ma-287	279	23	1)−	1)−	PROPN
ma-287	279	24	sin[log(x	sin[log(x	NOUN
ma-287	279	25	)	)	PUNCT
ma-287	279	26	]	]	PUNCT
ma-287	280	1	=	=	PUNCT
ma-287	280	2	φ(x	φ(x	PROPN
ma-287	280	3	−	−	PROPN
ma-287	280	4	1)−	1)−	NUM
ma-287	280	5	φ[log(x	φ[log(x	NUM
ma-287	280	6	)	)	PUNCT
ma-287	280	7	]	]	PUNCT
ma-287	281	1	=	=	PUNCT
ma-287	282	1	+	+	PUNCT
ma-287	282	2	∞∑	∞∑	NUM
ma-287	282	3	k=1	k=1	ADJ
ma-287	282	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	282	5	)	)	PUNCT
ma-287	282	6	,	,	PUNCT
ma-287	282	7	where	where	SCONJ
ma-287	282	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	282	9	)	)	PUNCT
ma-287	283	1	=	=	PUNCT
ma-287	283	2	φ	φ	PROPN
ma-287	283	3	[	[	PUNCT
ma-287	283	4	2k−1(x2	2k−1(x2	NOUN
ma-287	283	5	−(k−1	−(k−1	NOUN
ma-287	283	6	)	)	PUNCT
ma-287	284	1	−	−	PROPN
ma-287	284	2	1	1	NUM
ma-287	284	3	)	)	PUNCT
ma-287	284	4	]	]	PUNCT
ma-287	285	1	−	−	PROPN
ma-287	285	2	φ	φ	PROPN
ma-287	285	3	[	[	PUNCT
ma-287	285	4	2k(x2	2k(x2	NOUN
ma-287	285	5	−k	−k	ADJ
ma-287	285	6	−	−	NOUN
ma-287	285	7	1	1	NUM
ma-287	285	8	)	)	PUNCT
ma-287	285	9	]	]	PUNCT
ma-287	286	1	=	=	PUNCT
ma-287	286	2	sin	sin	NOUN
ma-287	286	3	[	[	PUNCT
ma-287	286	4	2k−1(x2	2k−1(x2	NOUN
ma-287	286	5	−(k−1	−(k−1	NOUN
ma-287	286	6	)	)	PUNCT
ma-287	286	7	−	−	PROPN
ma-287	286	8	1	1	NUM
ma-287	286	9	)	)	PUNCT
ma-287	286	10	]	]	PUNCT
ma-287	287	1	−	−	PROPN
ma-287	287	2	sin	sin	NOUN
ma-287	287	3	[	[	PUNCT
ma-287	287	4	2k(x2	2k(x2	NOUN
ma-287	287	5	−k	−k	ADJ
ma-287	287	6	−	−	NOUN
ma-287	287	7	1	1	NUM
ma-287	287	8	)	)	PUNCT
ma-287	287	9	]	]	PUNCT
ma-287	287	10	.	.	PUNCT
ma-287	288	1	using	use	VERB
ma-287	288	2	the	the	DET
ma-287	288	3	standard	standard	ADJ
ma-287	288	4	trigonometric	trigonometric	ADJ
ma-287	288	5	formula	formula	NOUN
ma-287	288	6	sin(t)−	sin(t)−	PROPN
ma-287	288	7	sin(u	sin(u	PROPN
ma-287	288	8	)	)	PUNCT
ma-287	289	1	=	=	SYM
ma-287	289	2	2	2	NUM
ma-287	289	3	sin[(t	sin[(t	NOUN
ma-287	289	4	−	−	PROPN
ma-287	289	5	u)/2	u)/2	PROPN
ma-287	289	6	]	]	PUNCT
ma-287	289	7	cos[(t	cos[(t	PROPN
ma-287	289	8	+	+	CCONJ
ma-287	289	9	u)/2],we	u)/2],we	NOUN
ma-287	289	10	get	get	VERB
ma-287	289	11	αk(φ)(x	αk(φ)(x	NUM
ma-287	289	12	)	)	PUNCT
ma-287	290	1	=	=	SYM
ma-287	290	2	2	2	NUM
ma-287	290	3	sin	sin	NOUN
ma-287	290	4	[	[	PUNCT
ma-287	290	5	2k−2(x2	2k−2(x2	NOUN
ma-287	290	6	−(k−1	−(k−1	NOUN
ma-287	290	7	)	)	PUNCT
ma-287	291	1	−	−	PROPN
ma-287	291	2	1)−	1)−	NUM
ma-287	291	3	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	291	4	−	−	PROPN
ma-287	291	5	1	1	NUM
ma-287	291	6	)	)	PUNCT
ma-287	291	7	]	]	PUNCT
ma-287	292	1	cos	cos	PROPN
ma-287	292	2	[	[	X
ma-287	292	3	2k−2(x2	2k−2(x2	NOUN
ma-287	292	4	−(k−1	−(k−1	NOUN
ma-287	292	5	)	)	PUNCT
ma-287	292	6	−	−	ADP
ma-287	293	1	1	1	X
ma-287	293	2	)	)	PUNCT
ma-287	293	3	+	+	CCONJ
ma-287	293	4	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	293	5	−	−	ADP
ma-287	293	6	1	1	NUM
ma-287	293	7	)	)	PUNCT
ma-287	293	8	]	]	PUNCT
ma-287	293	9	=	=	SYM
ma-287	293	10	2	2	NUM
ma-287	293	11	sin	sin	NOUN
ma-287	293	12	[	[	PUNCT
ma-287	293	13	2k−2(x2	2k−2(x2	NOUN
ma-287	293	14	−k	−k	NOUN
ma-287	293	15	−	−	PROPN
ma-287	293	16	1)2	1)2	NUM
ma-287	293	17	]	]	PUNCT
ma-287	294	1	cos	cos	PROPN
ma-287	294	2	[	[	PUNCT
ma-287	294	3	2k−2(x2	2k−2(x2	NOUN
ma-287	294	4	−k	−k	NOUN
ma-287	294	5	−	−	PROPN
ma-287	294	6	1)(3	1)(3	PROPN
ma-287	294	7	+	+	CCONJ
ma-287	294	8	x2−k	x2−k	PROPN
ma-287	294	9	)	)	PUNCT
ma-287	294	10	]	]	PUNCT
ma-287	294	11	.	.	PUNCT
ma-287	295	1	hence	hence	ADV
ma-287	295	2	,	,	PUNCT
ma-287	295	3	we	we	PRON
ma-287	295	4	have	have	VERB
ma-287	295	5	sin(x	sin(x	PROPN
ma-287	295	6	−	−	PROPN
ma-287	295	7	1)−	1)−	PROPN
ma-287	295	8	sin[log(x	sin[log(x	NOUN
ma-287	295	9	)	)	PUNCT
ma-287	295	10	]	]	PUNCT
ma-287	296	1	=	=	PUNCT
ma-287	296	2	2	2	NUM
ma-287	296	3	+	+	ADP
ma-287	296	4	∞∑	∞∑	NUM
ma-287	296	5	k=1	k=1	PRON
ma-287	296	6	sin	sin	NOUN
ma-287	296	7	[	[	PUNCT
ma-287	296	8	2k−2(x2	2k−2(x2	NOUN
ma-287	296	9	−k	−k	NOUN
ma-287	296	10	−	−	PROPN
ma-287	296	11	1)2	1)2	NUM
ma-287	296	12	]	]	PUNCT
ma-287	297	1	cos	cos	PROPN
ma-287	297	2	[	[	PUNCT
ma-287	297	3	2k−2(x2	2k−2(x2	NOUN
ma-287	297	4	−k	−k	NOUN
ma-287	297	5	−	−	PROPN
ma-287	297	6	1)(3	1)(3	PROPN
ma-287	297	7	+	+	CCONJ
ma-287	297	8	x2−k	x2−k	PROPN
ma-287	297	9	)	)	PUNCT
ma-287	297	10	]	]	PUNCT
ma-287	297	11	.	.	PUNCT
ma-287	298	1	(	(	PUNCT
ma-287	298	2	8)	8)	NUM
ma-287	298	3	for	for	ADP
ma-287	298	4	x	x	SYM
ma-287	298	5	>	>	X
ma-287	298	6	0	0	NUM
ma-287	298	7	,	,	PUNCT
ma-287	298	8	it	it	PRON
ma-287	298	9	follows	follow	VERB
ma-287	298	10	from	from	ADP
ma-287	298	11	theorem	theorem	ADJ
ma-287	298	12	2.1	2.1	NUM
ma-287	298	13	with	with	ADP
ma-287	298	14	φ(t	φ(t	NOUN
ma-287	298	15	)	)	PUNCT
ma-287	299	1	=	=	SYM
ma-287	299	2	cos(t	cos(t	PROPN
ma-287	299	3	)	)	PUNCT
ma-287	299	4	that	that	SCONJ
ma-287	299	5	cos(x	cos(x	PROPN
ma-287	299	6	−	−	PROPN
ma-287	299	7	1)−	1)−	NUM
ma-287	299	8	cos[log(x	cos[log(x	NOUN
ma-287	299	9	)	)	PUNCT
ma-287	299	10	]	]	PUNCT
ma-287	300	1	=	=	PUNCT
ma-287	300	2	φ(x	φ(x	PROPN
ma-287	300	3	−	−	PROPN
ma-287	300	4	1)−	1)−	NUM
ma-287	300	5	φ[log(x	φ[log(x	NUM
ma-287	300	6	)	)	PUNCT
ma-287	300	7	]	]	PUNCT
ma-287	301	1	=	=	PUNCT
ma-287	302	1	+	+	PUNCT
ma-287	302	2	∞∑	∞∑	NUM
ma-287	302	3	k=1	k=1	ADJ
ma-287	302	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	302	5	)	)	PUNCT
ma-287	302	6	,	,	PUNCT
ma-287	302	7	where	where	SCONJ
ma-287	302	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	302	9	)	)	PUNCT
ma-287	303	1	=	=	PUNCT
ma-287	303	2	φ	φ	PROPN
ma-287	303	3	[	[	PUNCT
ma-287	303	4	2k−1(x2	2k−1(x2	NOUN
ma-287	303	5	−(k−1	−(k−1	NOUN
ma-287	303	6	)	)	PUNCT
ma-287	304	1	−	−	PROPN
ma-287	304	2	1	1	NUM
ma-287	304	3	)	)	PUNCT
ma-287	304	4	]	]	PUNCT
ma-287	305	1	−	−	PROPN
ma-287	305	2	φ	φ	PROPN
ma-287	305	3	[	[	PUNCT
ma-287	305	4	2k(x2	2k(x2	NOUN
ma-287	305	5	−k	−k	ADJ
ma-287	305	6	−	−	NOUN
ma-287	305	7	1	1	NUM
ma-287	305	8	)	)	PUNCT
ma-287	305	9	]	]	PUNCT
ma-287	306	1	=	=	PUNCT
ma-287	306	2	cos	cos	X
ma-287	306	3	[	[	PUNCT
ma-287	306	4	2k−1(x2	2k−1(x2	NOUN
ma-287	306	5	−(k−1	−(k−1	NOUN
ma-287	306	6	)	)	PUNCT
ma-287	306	7	−	−	PROPN
ma-287	306	8	1	1	NUM
ma-287	306	9	)	)	PUNCT
ma-287	306	10	]	]	PUNCT
ma-287	307	1	−	−	PROPN
ma-287	307	2	cos	cos	X
ma-287	307	3	[	[	PUNCT
ma-287	307	4	2k(x2	2k(x2	NUM
ma-287	307	5	−k	−k	ADJ
ma-287	307	6	−	−	NOUN
ma-287	307	7	1	1	NUM
ma-287	307	8	)	)	PUNCT
ma-287	307	9	]	]	PUNCT
ma-287	307	10	.	.	PUNCT
ma-287	308	1	using	use	VERB
ma-287	308	2	the	the	DET
ma-287	308	3	standard	standard	ADJ
ma-287	308	4	trigonometric	trigonometric	ADJ
ma-287	308	5	formula	formula	NOUN
ma-287	308	6	cos(t)−cos(u	cos(t)−cos(u	NOUN
ma-287	308	7	)	)	PUNCT
ma-287	308	8	=	=	SYM
ma-287	308	9	−2	−2	PROPN
ma-287	308	10	sin[(t−u)/2	sin[(t−u)/2	PROPN
ma-287	308	11	]	]	X
ma-287	309	1	sin[(t+u)/2],we	sin[(t+u)/2],we	PROPN
ma-287	309	2	obtain	obtain	VERB
ma-287	309	3	αk(φ)(x	αk(φ)(x	NUM
ma-287	309	4	)	)	PUNCT
ma-287	310	1	=	=	SYM
ma-287	310	2	−2	−2	PROPN
ma-287	310	3	sin	sin	NOUN
ma-287	310	4	[	[	PUNCT
ma-287	310	5	2k−2(x2	2k−2(x2	NOUN
ma-287	310	6	−(k−1	−(k−1	NOUN
ma-287	310	7	)	)	PUNCT
ma-287	311	1	−	−	PROPN
ma-287	311	2	1)−	1)−	NUM
ma-287	311	3	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	311	4	−	−	NOUN
ma-287	311	5	1	1	NUM
ma-287	311	6	)	)	PUNCT
ma-287	312	1	]	]	PUNCT
ma-287	312	2	sin	sin	NOUN
ma-287	312	3	[	[	PUNCT
ma-287	312	4	2k−2(x2	2k−2(x2	NOUN
ma-287	312	5	−(k−1	−(k−1	NOUN
ma-287	312	6	)	)	PUNCT
ma-287	313	1	−	−	ADP
ma-287	313	2	1	1	X
ma-287	313	3	)	)	PUNCT
ma-287	313	4	+	+	CCONJ
ma-287	313	5	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	313	6	−	−	ADP
ma-287	313	7	1	1	NUM
ma-287	313	8	)	)	PUNCT
ma-287	313	9	]	]	PUNCT
ma-287	314	1	=	=	PUNCT
ma-287	314	2	−2	−2	PROPN
ma-287	314	3	sin	sin	NOUN
ma-287	314	4	[	[	PUNCT
ma-287	314	5	2k−2(x2	2k−2(x2	NOUN
ma-287	314	6	−k	−k	NOUN
ma-287	314	7	−	−	PROPN
ma-287	314	8	1)2	1)2	NUM
ma-287	314	9	]	]	PUNCT
ma-287	314	10	sin	sin	NOUN
ma-287	314	11	[	[	PUNCT
ma-287	314	12	2k−2(x2	2k−2(x2	NOUN
ma-287	314	13	−k	−k	NOUN
ma-287	314	14	−	−	PROPN
ma-287	314	15	1)(3	1)(3	PROPN
ma-287	314	16	+	+	CCONJ
ma-287	314	17	x2−k	x2−k	PROPN
ma-287	314	18	)	)	PUNCT
ma-287	314	19	]	]	PUNCT
ma-287	314	20	.	.	PUNCT
ma-287	315	1	hence	hence	ADV
ma-287	315	2	,	,	PUNCT
ma-287	315	3	we	we	PRON
ma-287	315	4	have	have	VERB
ma-287	315	5	cos(x	cos(x	PROPN
ma-287	315	6	−	−	PROPN
ma-287	315	7	1)−	1)−	NUM
ma-287	315	8	cos[log(x	cos[log(x	NOUN
ma-287	315	9	)	)	PUNCT
ma-287	315	10	]	]	PUNCT
ma-287	316	1	=	=	PUNCT
ma-287	316	2	−2	−2	NOUN
ma-287	317	1	+	+	NOUN
ma-287	317	2	∞∑	∞∑	ADJ
ma-287	317	3	k=1	k=1	PUNCT
ma-287	317	4	sin	sin	NOUN
ma-287	317	5	[	[	PUNCT
ma-287	317	6	2k−2(x2	2k−2(x2	NOUN
ma-287	317	7	−k	−k	NOUN
ma-287	317	8	−	−	PROPN
ma-287	317	9	1)2	1)2	NUM
ma-287	317	10	]	]	PUNCT
ma-287	317	11	sin	sin	NOUN
ma-287	317	12	[	[	PUNCT
ma-287	317	13	2k−2(x2	2k−2(x2	NOUN
ma-287	317	14	−k	−k	NOUN
ma-287	317	15	−	−	PROPN
ma-287	317	16	1)(3	1)(3	PROPN
ma-287	317	17	+	+	CCONJ
ma-287	317	18	x2−k	x2−k	PROPN
ma-287	317	19	)	)	PUNCT
ma-287	317	20	]	]	PUNCT
ma-287	317	21	.	.	PUNCT
ma-287	318	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	318	2	eur	eur	PROPN
ma-287	318	3	.	.	PUNCT
ma-287	319	1	j.	j.	PROPN
ma-287	319	2	math	math	PROPN
ma-287	319	3	.	.	PUNCT
ma-287	320	1	anal	anal	PROPN
ma-287	320	2	.	.	PUNCT
ma-287	321	1	10.28924	10.28924	NUM
ma-287	321	2	/	/	SYM
ma-287	321	3	ada	ada	PROPN
ma-287	321	4	/	/	SYM
ma-287	321	5	ma.5.12	ma.5.12	PROPN
ma-287	321	6	9(9	9(9	NUM
ma-287	321	7	)	)	PUNCT
ma-287	321	8	for	for	ADP
ma-287	321	9	x	x	SYM
ma-287	321	10	>	>	X
ma-287	321	11	0	0	NUM
ma-287	321	12	,	,	PUNCT
ma-287	321	13	applying	apply	VERB
ma-287	321	14	theorem	theorem	NOUN
ma-287	321	15	2.1	2.1	NUM
ma-287	321	16	to	to	ADP
ma-287	321	17	φ(t	φ(t	PROPN
ma-287	321	18	)	)	PUNCT
ma-287	321	19	=	=	SYM
ma-287	321	20	sinh(t	sinh(t	PROPN
ma-287	321	21	)	)	PUNCT
ma-287	321	22	,	,	PUNCT
ma-287	321	23	we	we	PRON
ma-287	321	24	establish	establish	VERB
ma-287	321	25	that	that	SCONJ
ma-287	321	26	sinh(x	sinh(x	PROPN
ma-287	321	27	−	−	PROPN
ma-287	321	28	1)−	1)−	PROPN
ma-287	321	29	sinh[log(x	sinh[log(x	PROPN
ma-287	321	30	)	)	PUNCT
ma-287	321	31	]	]	PUNCT
ma-287	322	1	=	=	PUNCT
ma-287	322	2	φ(x	φ(x	PROPN
ma-287	322	3	−	−	PROPN
ma-287	322	4	1)−	1)−	NUM
ma-287	322	5	φ[log(x	φ[log(x	NUM
ma-287	322	6	)	)	PUNCT
ma-287	322	7	]	]	PUNCT
ma-287	323	1	=	=	PUNCT
ma-287	324	1	+	+	PUNCT
ma-287	324	2	∞∑	∞∑	NUM
ma-287	324	3	k=1	k=1	ADJ
ma-287	324	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	324	5	)	)	PUNCT
ma-287	324	6	,	,	PUNCT
ma-287	324	7	where	where	SCONJ
ma-287	324	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	324	9	)	)	PUNCT
ma-287	325	1	=	=	PUNCT
ma-287	325	2	φ	φ	PROPN
ma-287	325	3	[	[	PUNCT
ma-287	325	4	2k−1(x2	2k−1(x2	NOUN
ma-287	325	5	−(k−1	−(k−1	NOUN
ma-287	325	6	)	)	PUNCT
ma-287	326	1	−	−	PROPN
ma-287	326	2	1	1	NUM
ma-287	326	3	)	)	PUNCT
ma-287	326	4	]	]	PUNCT
ma-287	327	1	−	−	PROPN
ma-287	327	2	φ	φ	PROPN
ma-287	327	3	[	[	PUNCT
ma-287	327	4	2k(x2	2k(x2	NOUN
ma-287	327	5	−k	−k	ADJ
ma-287	327	6	−	−	NOUN
ma-287	327	7	1	1	NUM
ma-287	327	8	)	)	PUNCT
ma-287	327	9	]	]	PUNCT
ma-287	328	1	=	=	PUNCT
ma-287	328	2	sinh	sinh	NOUN
ma-287	328	3	[	[	PUNCT
ma-287	328	4	2k−1(x2	2k−1(x2	NOUN
ma-287	328	5	−(k−1	−(k−1	NOUN
ma-287	328	6	)	)	PUNCT
ma-287	328	7	−	−	PROPN
ma-287	328	8	1	1	NUM
ma-287	328	9	)	)	PUNCT
ma-287	328	10	]	]	PUNCT
ma-287	328	11	−	−	PROPN
ma-287	328	12	sinh	sinh	NOUN
ma-287	328	13	[	[	PUNCT
ma-287	328	14	2k(x2	2k(x2	NOUN
ma-287	328	15	−k	−k	ADJ
ma-287	328	16	−	−	NOUN
ma-287	328	17	1	1	NUM
ma-287	328	18	)	)	PUNCT
ma-287	328	19	]	]	PUNCT
ma-287	328	20	.	.	PUNCT
ma-287	329	1	using	use	VERB
ma-287	329	2	the	the	DET
ma-287	329	3	standard	standard	ADJ
ma-287	329	4	hyperbolic	hyperbolic	ADJ
ma-287	329	5	formula	formula	NOUN
ma-287	329	6	sinh(t)−	sinh(t)−	PROPN
ma-287	329	7	sinh(u	sinh(u	PROPN
ma-287	329	8	)	)	PUNCT
ma-287	329	9	=	=	SYM
ma-287	329	10	2	2	NUM
ma-287	329	11	sinh[(t−u)/2	sinh[(t−u)/2	NOUN
ma-287	329	12	]	]	X
ma-287	329	13	cosh[(t+u)/2],we	cosh[(t+u)/2],we	PROPN
ma-287	329	14	get	get	VERB
ma-287	329	15	αk(φ)(x	αk(φ)(x	PROPN
ma-287	329	16	)	)	PUNCT
ma-287	330	1	=	=	SYM
ma-287	330	2	2	2	NUM
ma-287	330	3	sinh	sinh	NOUN
ma-287	330	4	[	[	PUNCT
ma-287	330	5	2k−2(x2	2k−2(x2	NOUN
ma-287	330	6	−(k−1	−(k−1	NOUN
ma-287	330	7	)	)	PUNCT
ma-287	331	1	−	−	PROPN
ma-287	331	2	1)−	1)−	NUM
ma-287	331	3	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	331	4	−	−	NOUN
ma-287	331	5	1	1	NUM
ma-287	331	6	)	)	PUNCT
ma-287	331	7	]	]	PUNCT
ma-287	332	1	cosh	cosh	PROPN
ma-287	332	2	[	[	PUNCT
ma-287	332	3	2k−2(x2	2k−2(x2	NOUN
ma-287	332	4	−(k−1	−(k−1	NOUN
ma-287	332	5	)	)	PUNCT
ma-287	332	6	−	−	ADP
ma-287	332	7	1	1	X
ma-287	332	8	)	)	PUNCT
ma-287	332	9	+	+	CCONJ
ma-287	332	10	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	332	11	−	−	ADP
ma-287	332	12	1	1	NUM
ma-287	332	13	)	)	PUNCT
ma-287	332	14	]	]	PUNCT
ma-287	333	1	=	=	SYM
ma-287	333	2	2	2	NUM
ma-287	333	3	sinh	sinh	NOUN
ma-287	333	4	[	[	PUNCT
ma-287	333	5	2k−2(x2	2k−2(x2	NOUN
ma-287	333	6	−k	−k	NOUN
ma-287	333	7	−	−	PROPN
ma-287	333	8	1)2	1)2	NUM
ma-287	333	9	]	]	PUNCT
ma-287	333	10	cosh	cosh	PROPN
ma-287	333	11	[	[	PUNCT
ma-287	333	12	2k−2(x2	2k−2(x2	NOUN
ma-287	333	13	−k	−k	NOUN
ma-287	333	14	−	−	PROPN
ma-287	333	15	1)(3	1)(3	PROPN
ma-287	333	16	+	+	CCONJ
ma-287	333	17	x2−k	x2−k	PROPN
ma-287	333	18	)	)	PUNCT
ma-287	333	19	]	]	PUNCT
ma-287	333	20	.	.	PUNCT
ma-287	334	1	hence	hence	ADV
ma-287	334	2	,	,	PUNCT
ma-287	334	3	we	we	PRON
ma-287	334	4	have	have	VERB
ma-287	334	5	sinh(x	sinh(x	NUM
ma-287	334	6	−	−	PROPN
ma-287	334	7	1)−	1)−	PROPN
ma-287	334	8	sinh[log(x	sinh[log(x	PROPN
ma-287	334	9	)	)	PUNCT
ma-287	334	10	]	]	PUNCT
ma-287	335	1	=	=	PUNCT
ma-287	335	2	2	2	NUM
ma-287	335	3	+	+	ADP
ma-287	335	4	∞∑	∞∑	NUM
ma-287	335	5	k=1	k=1	PROPN
ma-287	335	6	sinh	sinh	NOUN
ma-287	335	7	[	[	PUNCT
ma-287	335	8	2k−2(x2	2k−2(x2	NOUN
ma-287	335	9	−k	−k	NOUN
ma-287	335	10	−	−	PROPN
ma-287	335	11	1)2	1)2	NUM
ma-287	335	12	]	]	PUNCT
ma-287	335	13	cosh	cosh	PROPN
ma-287	335	14	[	[	PUNCT
ma-287	335	15	2k−2(x2	2k−2(x2	NOUN
ma-287	335	16	−k	−k	NOUN
ma-287	335	17	−	−	PROPN
ma-287	335	18	1)(3	1)(3	PROPN
ma-287	335	19	+	+	CCONJ
ma-287	335	20	x2−k	x2−k	PROPN
ma-287	335	21	)	)	PUNCT
ma-287	335	22	]	]	PUNCT
ma-287	335	23	.	.	PUNCT
ma-287	336	1	(	(	PUNCT
ma-287	336	2	10	10	NUM
ma-287	336	3	)	)	PUNCT
ma-287	336	4	for	for	ADP
ma-287	336	5	x	x	SYM
ma-287	336	6	>	>	X
ma-287	336	7	0	0	NUM
ma-287	336	8	,	,	PUNCT
ma-287	336	9	using	use	VERB
ma-287	336	10	theorem	theorem	NOUN
ma-287	336	11	2.1	2.1	NUM
ma-287	336	12	with	with	ADP
ma-287	336	13	φ(t	φ(t	NOUN
ma-287	336	14	)	)	PUNCT
ma-287	337	1	=	=	SYM
ma-287	337	2	cosh(t	cosh(t	PROPN
ma-287	337	3	)	)	PUNCT
ma-287	338	1	,	,	PUNCT
ma-287	338	2	we	we	PRON
ma-287	338	3	find	find	VERB
ma-287	338	4	that	that	SCONJ
ma-287	338	5	cosh(x	cosh(x	NOUN
ma-287	338	6	−	−	PROPN
ma-287	338	7	1)−	1)−	PROPN
ma-287	338	8	cosh[log(x	cosh[log(x	NOUN
ma-287	338	9	)	)	PUNCT
ma-287	338	10	]	]	PUNCT
ma-287	339	1	=	=	PUNCT
ma-287	339	2	φ(x	φ(x	PROPN
ma-287	339	3	−	−	PROPN
ma-287	339	4	1)−	1)−	NUM
ma-287	339	5	φ[log(x	φ[log(x	NUM
ma-287	339	6	)	)	PUNCT
ma-287	339	7	]	]	PUNCT
ma-287	340	1	=	=	PUNCT
ma-287	341	1	+	+	PUNCT
ma-287	341	2	∞∑	∞∑	NUM
ma-287	341	3	k=1	k=1	ADJ
ma-287	341	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	341	5	)	)	PUNCT
ma-287	341	6	,	,	PUNCT
ma-287	341	7	where	where	SCONJ
ma-287	341	8	αk(φ)(x	αk(φ)(x	NOUN
ma-287	341	9	)	)	PUNCT
ma-287	342	1	=	=	PUNCT
ma-287	342	2	φ	φ	PROPN
ma-287	342	3	[	[	PUNCT
ma-287	342	4	2k−1(x2	2k−1(x2	NOUN
ma-287	342	5	−(k−1	−(k−1	NOUN
ma-287	342	6	)	)	PUNCT
ma-287	343	1	−	−	PROPN
ma-287	343	2	1	1	NUM
ma-287	343	3	)	)	PUNCT
ma-287	343	4	]	]	PUNCT
ma-287	344	1	−	−	PROPN
ma-287	344	2	φ	φ	PROPN
ma-287	344	3	[	[	PUNCT
ma-287	344	4	2k(x2	2k(x2	NOUN
ma-287	344	5	−k	−k	ADJ
ma-287	344	6	−	−	NOUN
ma-287	344	7	1	1	NUM
ma-287	344	8	)	)	PUNCT
ma-287	344	9	]	]	PUNCT
ma-287	345	1	=	=	PUNCT
ma-287	345	2	cosh	cosh	NOUN
ma-287	345	3	[	[	PUNCT
ma-287	345	4	2k−1(x2	2k−1(x2	NOUN
ma-287	345	5	−(k−1	−(k−1	NOUN
ma-287	345	6	)	)	PUNCT
ma-287	345	7	−	−	PROPN
ma-287	345	8	1	1	NUM
ma-287	345	9	)	)	PUNCT
ma-287	345	10	]	]	PUNCT
ma-287	346	1	−	−	PROPN
ma-287	346	2	cosh	cosh	PROPN
ma-287	346	3	[	[	PUNCT
ma-287	346	4	2k(x2	2k(x2	NOUN
ma-287	346	5	−k	−k	ADJ
ma-287	346	6	−	−	NOUN
ma-287	346	7	1	1	NUM
ma-287	346	8	)	)	PUNCT
ma-287	346	9	]	]	PUNCT
ma-287	346	10	.	.	PUNCT
ma-287	347	1	using	use	VERB
ma-287	347	2	the	the	DET
ma-287	347	3	standard	standard	ADJ
ma-287	347	4	trigonometric	trigonometric	ADJ
ma-287	347	5	formula	formula	NOUN
ma-287	347	6	cosh(t	cosh(t	NOUN
ma-287	347	7	)	)	PUNCT
ma-287	348	1	−	−	PROPN
ma-287	348	2	cosh(u	cosh(u	NOUN
ma-287	348	3	)	)	PUNCT
ma-287	348	4	=	=	SYM
ma-287	348	5	2	2	NUM
ma-287	348	6	sinh[(t	sinh[(t	NOUN
ma-287	348	7	−	−	PROPN
ma-287	348	8	u)/2	u)/2	PROPN
ma-287	348	9	]	]	PUNCT
ma-287	348	10	sinh[(t	sinh[(t	NOUN
ma-287	348	11	+	+	CCONJ
ma-287	349	1	u)/2	u)/2	PROPN
ma-287	349	2	]	]	PUNCT
ma-287	349	3	,	,	PUNCT
ma-287	349	4	we	we	PRON
ma-287	349	5	obtain	obtain	VERB
ma-287	349	6	αk(φ)(x	αk(φ)(x	NUM
ma-287	349	7	)	)	PUNCT
ma-287	350	1	=	=	SYM
ma-287	350	2	2	2	NUM
ma-287	350	3	sinh	sinh	NOUN
ma-287	350	4	[	[	PUNCT
ma-287	350	5	2k−2(x2	2k−2(x2	NOUN
ma-287	350	6	−(k−1	−(k−1	NOUN
ma-287	350	7	)	)	PUNCT
ma-287	351	1	−	−	PROPN
ma-287	351	2	1)−	1)−	NUM
ma-287	351	3	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	351	4	−	−	NOUN
ma-287	351	5	1	1	NUM
ma-287	351	6	)	)	PUNCT
ma-287	352	1	]	]	PUNCT
ma-287	352	2	sinh	sinh	PROPN
ma-287	352	3	[	[	PUNCT
ma-287	352	4	2k−2(x2	2k−2(x2	NOUN
ma-287	352	5	−(k−1	−(k−1	NOUN
ma-287	352	6	)	)	PUNCT
ma-287	353	1	−	−	ADP
ma-287	353	2	1	1	X
ma-287	353	3	)	)	PUNCT
ma-287	353	4	+	+	CCONJ
ma-287	353	5	2k−1(x2−k	2k−1(x2−k	NUM
ma-287	353	6	−	−	ADP
ma-287	353	7	1	1	NUM
ma-287	353	8	)	)	PUNCT
ma-287	353	9	]	]	PUNCT
ma-287	354	1	=	=	SYM
ma-287	354	2	2	2	NUM
ma-287	354	3	sinh	sinh	NOUN
ma-287	354	4	[	[	PUNCT
ma-287	354	5	2k−2(x2	2k−2(x2	NOUN
ma-287	354	6	−k	−k	NOUN
ma-287	354	7	−	−	PROPN
ma-287	354	8	1)2	1)2	NUM
ma-287	354	9	]	]	PUNCT
ma-287	354	10	sinh	sinh	NOUN
ma-287	354	11	[	[	PUNCT
ma-287	354	12	2k−2(x2	2k−2(x2	NOUN
ma-287	354	13	−k	−k	NOUN
ma-287	354	14	−	−	PROPN
ma-287	354	15	1)(3	1)(3	PROPN
ma-287	354	16	+	+	CCONJ
ma-287	354	17	x2−k	x2−k	PROPN
ma-287	354	18	)	)	PUNCT
ma-287	354	19	]	]	PUNCT
ma-287	354	20	.	.	PUNCT
ma-287	355	1	hence	hence	ADV
ma-287	355	2	,	,	PUNCT
ma-287	355	3	we	we	PRON
ma-287	355	4	have	have	VERB
ma-287	355	5	cosh(x	cosh(x	NUM
ma-287	355	6	−	−	PROPN
ma-287	355	7	1)−	1)−	PROPN
ma-287	355	8	cosh[log(x	cosh[log(x	NOUN
ma-287	355	9	)	)	PUNCT
ma-287	355	10	]	]	PUNCT
ma-287	356	1	=	=	PUNCT
ma-287	356	2	2	2	NUM
ma-287	356	3	+	+	ADP
ma-287	356	4	∞∑	∞∑	NUM
ma-287	356	5	k=1	k=1	PROPN
ma-287	356	6	sinh	sinh	NOUN
ma-287	356	7	[	[	PUNCT
ma-287	356	8	2k−2(x2	2k−2(x2	NOUN
ma-287	356	9	−k	−k	NOUN
ma-287	356	10	−	−	PROPN
ma-287	356	11	1)2	1)2	NUM
ma-287	356	12	]	]	PUNCT
ma-287	356	13	sinh	sinh	NOUN
ma-287	356	14	[	[	PUNCT
ma-287	356	15	2k−2(x2	2k−2(x2	NOUN
ma-287	356	16	−k	−k	NOUN
ma-287	356	17	−	−	PROPN
ma-287	356	18	1)(3	1)(3	PROPN
ma-287	356	19	+	+	CCONJ
ma-287	356	20	x2−k	x2−k	PROPN
ma-287	356	21	)	)	PUNCT
ma-287	356	22	]	]	PUNCT
ma-287	356	23	.	.	PUNCT
ma-287	357	1	all	all	DET
ma-287	357	2	the	the	DET
ma-287	357	3	claimed	claim	VERB
ma-287	357	4	expansions	expansion	NOUN
ma-287	357	5	are	be	AUX
ma-287	357	6	established	establish	VERB
ma-287	357	7	,	,	PUNCT
ma-287	357	8	ending	end	VERB
ma-287	357	9	the	the	DET
ma-287	357	10	proof	proof	NOUN
ma-287	357	11	.	.	PUNCT
ma-287	358	1	�	�	PROPN
ma-287	358	2	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	358	3	eur	eur	PROPN
ma-287	358	4	.	.	PUNCT
ma-287	359	1	j.	j.	PROPN
ma-287	359	2	math	math	PROPN
ma-287	359	3	.	.	PUNCT
ma-287	360	1	anal	anal	PROPN
ma-287	360	2	.	.	PUNCT
ma-287	361	1	10.28924	10.28924	NUM
ma-287	361	2	/	/	SYM
ma-287	361	3	ada	ada	PROPN
ma-287	361	4	/	/	SYM
ma-287	361	5	ma.5.12	ma.5.12	PROPN
ma-287	361	6	10thus	10thus	NUM
ma-287	361	7	,	,	PUNCT
ma-287	361	8	proposition	proposition	NOUN
ma-287	361	9	2.2	2.2	NUM
ma-287	361	10	is	be	AUX
ma-287	361	11	derived	derive	VERB
ma-287	361	12	from	from	ADP
ma-287	361	13	theorem	theorem	ADJ
ma-287	361	14	2.1	2.1	NUM
ma-287	361	15	by	by	ADP
ma-287	361	16	using	use	VERB
ma-287	361	17	various	various	ADJ
ma-287	361	18	functions	function	NOUN
ma-287	361	19	φ	φ	NUM
ma-287	361	20	,	,	PUNCT
ma-287	361	21	namely	namely	ADV
ma-287	361	22	φ(t	φ(t	PROPN
ma-287	361	23	)	)	PUNCT
ma-287	361	24	=	=	SYM
ma-287	361	25	t	t	PROPN
ma-287	361	26	,	,	PUNCT
ma-287	361	27	φ(t	φ(t	PROPN
ma-287	361	28	)	)	PUNCT
ma-287	361	29	=	=	SYM
ma-287	361	30	1	1	NUM
ma-287	361	31	/	/	SYM
ma-287	361	32	t	t	NOUN
ma-287	361	33	,	,	PUNCT
ma-287	361	34	φ(t	φ(t	PROPN
ma-287	361	35	)	)	PUNCT
ma-287	361	36	=	=	SYM
ma-287	361	37	t2	t2	NOUN
ma-287	361	38	,	,	PUNCT
ma-287	361	39	φ(t	φ(t	PROPN
ma-287	361	40	)	)	PUNCT
ma-287	361	41	=	=	SYM
ma-287	362	1	√t	√t	NOUN
ma-287	362	2	,	,	PUNCT
ma-287	362	3	φ(t	φ(t	PROPN
ma-287	362	4	)	)	PUNCT
ma-287	362	5	=	=	SYM
ma-287	362	6	1/√t	1/√t	NUM
ma-287	362	7	,	,	PUNCT
ma-287	362	8	φ(t	φ(t	PROPN
ma-287	362	9	)	)	PUNCT
ma-287	362	10	=	=	SYM
ma-287	362	11	log(t	log(t	PROPN
ma-287	362	12	)	)	PUNCT
ma-287	362	13	,	,	PUNCT
ma-287	362	14	φ(t	φ(t	PROPN
ma-287	362	15	)	)	PUNCT
ma-287	362	16	=	=	SYM
ma-287	362	17	sin(t	sin(t	PROPN
ma-287	362	18	)	)	PUNCT
ma-287	362	19	,	,	PUNCT
ma-287	362	20	φ(t	φ(t	PROPN
ma-287	362	21	)	)	PUNCT
ma-287	362	22	=	=	SYM
ma-287	362	23	cos(t	cos(t	PROPN
ma-287	362	24	)	)	PUNCT
ma-287	362	25	,	,	PUNCT
ma-287	362	26	φ(t	φ(t	PROPN
ma-287	362	27	)	)	PUNCT
ma-287	362	28	=	=	SYM
ma-287	362	29	sinh(t	sinh(t	PROPN
ma-287	362	30	)	)	PUNCT
ma-287	362	31	and	and	CCONJ
ma-287	362	32	φ(t	φ(t	PROPN
ma-287	362	33	)	)	PUNCT
ma-287	362	34	=	=	SYM
ma-287	362	35	cosh(t	cosh(t	PROPN
ma-287	362	36	)	)	PUNCT
ma-287	362	37	,	,	PUNCT
ma-287	362	38	one	one	NUM
ma-287	362	39	for	for	ADP
ma-287	362	40	each	each	DET
ma-287	362	41	sub	sub	NOUN
ma-287	362	42	-	-	NOUN
ma-287	362	43	result	result	ADJ
ma-287	362	44	,	,	PUNCT
ma-287	362	45	in	in	ADP
ma-287	362	46	order	order	NOUN
ma-287	362	47	.	.	PUNCT
ma-287	363	1	in	in	ADP
ma-287	363	2	addition	addition	NOUN
ma-287	363	3	,	,	PUNCT
ma-287	363	4	some	some	DET
ma-287	363	5	extendedresults	extendedresult	NOUN
ma-287	363	6	can	can	AUX
ma-287	363	7	be	be	AUX
ma-287	363	8	proved	prove	VERB
ma-287	363	9	.	.	PUNCT
ma-287	364	1	for	for	ADP
ma-287	364	2	example	example	NOUN
ma-287	364	3	,	,	PUNCT
ma-287	364	4	for	for	ADP
ma-287	364	5	x	x	SYM
ma-287	364	6	>	>	X
ma-287	364	7	0	0	PROPN
ma-287	364	8	,	,	PUNCT
ma-287	364	9	based	base	VERB
ma-287	364	10	on	on	ADP
ma-287	364	11	the	the	DET
ma-287	364	12	proof	proof	NOUN
ma-287	364	13	theorem	theorem	VERB
ma-287	364	14	2.1	2.1	NUM
ma-287	364	15	with	with	ADP
ma-287	364	16	the	the	DET
ma-287	364	17	function	function	NOUN
ma-287	364	18	φ(t	φ(t	PROPN
ma-287	364	19	)	)	PUNCT
ma-287	364	20	=	=	SYM
ma-287	364	21	√	√	NUM
ma-287	364	22	|t|	|t|	PROPN
ma-287	364	23	,	,	PUNCT
ma-287	364	24	we	we	PRON
ma-287	364	25	can	can	AUX
ma-287	364	26	extend	extend	VERB
ma-287	364	27	the	the	DET
ma-287	364	28	item	item	NOUN
ma-287	364	29	numbered	number	VERB
ma-287	364	30	4	4	NUM
ma-287	364	31	as√	as√	NOUN
ma-287	364	32	|x	|x	NOUN
ma-287	364	33	−	−	PROPN
ma-287	364	34	1|	1|	NUM
ma-287	364	35	−	−	NOUN
ma-287	365	1	√	√	PROPN
ma-287	366	1	|	|	ADV
ma-287	366	2	log(x)|	log(x)|	PROPN
ma-287	367	1	=	=	NOUN
ma-287	367	2	sign(x	sign(x	ADP
ma-287	367	3	−	−	NOUN
ma-287	367	4	1	1	X
ma-287	367	5	)	)	PUNCT
ma-287	367	6	+	+	NOUN
ma-287	367	7	∞∑	∞∑	ADJ
ma-287	367	8	k=1	k=1	PUNCT
ma-287	368	1	2(k−1)/2|x2−k	2(k−1)/2|x2−k	NUM
ma-287	368	2	−	−	NOUN
ma-287	368	3	1|3/2√	1|3/2√	NUM
ma-287	368	4	1	1	NUM
ma-287	368	5	+	+	NUM
ma-287	368	6	x2	x2	NOUN
ma-287	368	7	−k	−k	PROPN
ma-287	368	8	+	+	CCONJ
ma-287	368	9	√	√	NUM
ma-287	368	10	2	2	NUM
ma-287	368	11	,	,	PUNCT
ma-287	368	12	where	where	SCONJ
ma-287	368	13	sign(x	sign(x	ADV
ma-287	368	14	−	−	PROPN
ma-287	368	15	1	1	NUM
ma-287	368	16	)	)	PUNCT
ma-287	368	17	=	=	SYM
ma-287	369	1			PROPN
ma-287	369	2	1	1	NUM
ma-287	370	1	if	if	SCONJ
ma-287	370	2	x	x	PROPN
ma-287	370	3	>	>	X
ma-287	370	4	1	1	NUM
ma-287	370	5	,	,	PUNCT
ma-287	370	6	0	0	PUNCT
ma-287	370	7	if	if	SCONJ
ma-287	370	8	x	x	X
ma-287	370	9	=	=	SYM
ma-287	370	10	1	1	NUM
ma-287	370	11	,	,	PUNCT
ma-287	370	12	−1	−1	VERB
ma-287	370	13	if	if	SCONJ
ma-287	370	14	x	x	X
ma-287	370	15	<	<	X
ma-287	370	16	1	1	X
ma-287	370	17	.	.	PUNCT
ma-287	370	18	analogous	analogous	ADJ
ma-287	370	19	extension	extension	NOUN
ma-287	370	20	of	of	ADP
ma-287	370	21	the	the	DET
ma-287	370	22	item	item	NOUN
ma-287	370	23	numbered	number	VERB
ma-287	370	24	5	5	NUM
ma-287	370	25	is	be	AUX
ma-287	370	26	possible	possible	ADJ
ma-287	370	27	.	.	PUNCT
ma-287	371	1	with	with	ADP
ma-287	371	2	a	a	DET
ma-287	371	3	little	little	ADJ
ma-287	371	4	mathematical	mathematical	ADJ
ma-287	371	5	effort	effort	NOUN
ma-287	371	6	,	,	PUNCT
ma-287	371	7	we	we	PRON
ma-287	371	8	can	can	AUX
ma-287	371	9	get	get	VERB
ma-287	371	10	similar	similar	ADJ
ma-287	371	11	series	series	NOUN
ma-287	371	12	expansions	expansion	NOUN
ma-287	371	13	by	by	ADP
ma-287	371	14	considering	consider	VERB
ma-287	371	15	the	the	DET
ma-287	371	16	trans	trans	ADJ
ma-287	371	17	-	-	ADJ
ma-287	371	18	lated	lated	ADJ
ma-287	371	19	version	version	NOUN
ma-287	371	20	of	of	ADP
ma-287	371	21	φ	φ	PROPN
ma-287	371	22	,	,	PUNCT
ma-287	371	23	i.e.	i.e.	X
ma-287	371	24	,	,	PUNCT
ma-287	371	25	φ(t	φ(t	PROPN
ma-287	371	26	;	;	PUNCT
ma-287	371	27	a	a	X
ma-287	371	28	)	)	PUNCT
ma-287	371	29	=	=	SYM
ma-287	371	30	φ(t	φ(t	PROPN
ma-287	371	31	+	+	CCONJ
ma-287	371	32	a	a	X
ma-287	371	33	)	)	PUNCT
ma-287	371	34	for	for	ADP
ma-287	371	35	some	some	PRON
ma-287	371	36	a	a	DET
ma-287	371	37	∈	∈	PROPN
ma-287	371	38	r.	r.	NOUN
ma-287	371	39	of	of	ADP
ma-287	371	40	course	course	NOUN
ma-287	371	41	,	,	PUNCT
ma-287	371	42	other	other	ADJ
ma-287	371	43	interesting	interesting	ADJ
ma-287	371	44	functionscan	functionscan	PROPN
ma-287	371	45	also	also	ADV
ma-287	371	46	be	be	AUX
ma-287	371	47	examined	examine	VERB
ma-287	371	48	for	for	ADP
ma-287	371	49	φ	φ	NUM
ma-287	371	50	,	,	PUNCT
ma-287	371	51	such	such	ADJ
ma-287	371	52	as	as	ADP
ma-287	371	53	φ(t	φ(t	PROPN
ma-287	371	54	)	)	PUNCT
ma-287	372	1	=	=	SYM
ma-287	372	2	arctanh(t	arctanh(t	PROPN
ma-287	372	3	)	)	PUNCT
ma-287	372	4	,	,	PUNCT
ma-287	372	5	which	which	PRON
ma-287	372	6	benefits	benefit	VERB
ma-287	372	7	from	from	ADP
ma-287	372	8	an	an	DET
ma-287	372	9	interesting	interesting	ADJ
ma-287	372	10	additionformula	additionformula	NOUN
ma-287	372	11	,	,	PUNCT
ma-287	372	12	among	among	ADP
ma-287	372	13	other	other	ADJ
ma-287	372	14	things.it	things.it	PROPN
ma-287	372	15	is	be	AUX
ma-287	372	16	important	important	ADJ
ma-287	372	17	to	to	PART
ma-287	372	18	note	note	VERB
ma-287	372	19	that	that	SCONJ
ma-287	372	20	the	the	DET
ma-287	372	21	convergence	convergence	NOUN
ma-287	372	22	of	of	ADP
ma-287	372	23	the	the	DET
ma-287	372	24	series	series	NOUN
ma-287	372	25	expansions	expansion	NOUN
ma-287	372	26	in	in	ADP
ma-287	372	27	proposition	proposition	NOUN
ma-287	372	28	2.2	2.2	NUM
ma-287	372	29	hasbeen	hasbeen	NOUN
ma-287	372	30	checked	check	VERB
ma-287	372	31	on	on	ADP
ma-287	372	32	the	the	DET
ma-287	372	33	basis	basis	NOUN
ma-287	372	34	of	of	ADP
ma-287	372	35	theoretical	theoretical	ADJ
ma-287	372	36	and	and	CCONJ
ma-287	372	37	practical	practical	ADJ
ma-287	372	38	work	work	NOUN
ma-287	372	39	.	.	PUNCT
ma-287	373	1	let	let	VERB
ma-287	373	2	us	we	PRON
ma-287	373	3	illustrate	illustrate	VERB
ma-287	373	4	graphically	graphically	ADV
ma-287	373	5	theconvergence	theconvergence	NOUN
ma-287	373	6	of	of	ADP
ma-287	373	7	the	the	DET
ma-287	373	8	series	series	NOUN
ma-287	373	9	expansion	expansion	NOUN
ma-287	373	10	in	in	ADP
ma-287	373	11	the	the	DET
ma-287	373	12	item	item	NOUN
ma-287	373	13	numbered	number	VERB
ma-287	373	14	3	3	NUM
ma-287	373	15	.	.	PUNCT
ma-287	373	16	to	to	PART
ma-287	373	17	do	do	VERB
ma-287	373	18	this	this	PRON
ma-287	373	19	,	,	PUNCT
ma-287	373	20	we	we	PRON
ma-287	373	21	consider	consider	VERB
ma-287	373	22	the	the	DET
ma-287	373	23	followingtruncated	followingtruncate	VERB
ma-287	373	24	-	-	PUNCT
ma-287	373	25	series	series	NOUN
ma-287	373	26	function	function	NOUN
ma-287	373	27	:	:	PUNCT
ma-287	373	28	ϕ(x	ϕ(x	X
ma-287	373	29	;	;	PUNCT
ma-287	373	30	m	m	X
ma-287	373	31	)	)	PUNCT
ma-287	373	32	=	=	SYM
ma-287	374	1	(	(	PUNCT
ma-287	374	2	x	x	SYM
ma-287	374	3	−	−	PROPN
ma-287	374	4	1)2	1)2	NUM
ma-287	374	5	−	−	PUNCT
ma-287	375	1	[	[	X
ma-287	375	2	log(x)]2	log(x)]2	INTJ
ma-287	375	3	−	−	NOUN
ma-287	375	4	m∑	m∑	ADP
ma-287	375	5	k=1	k=1	PROPN
ma-287	375	6	22(k−1)(x2	22(k−1)(x2	PROPN
ma-287	375	7	−k	−k	NOUN
ma-287	375	8	−	−	PROPN
ma-287	376	1	1)3(3	1)3(3	PROPN
ma-287	376	2	+	+	NUM
ma-287	376	3	x2−k	x2−k	PROPN
ma-287	376	4	)	)	PUNCT
ma-287	376	5	,	,	PUNCT
ma-287	376	6	where	where	SCONJ
ma-287	376	7	m	m	PRON
ma-287	376	8	denotes	denote	VERB
ma-287	376	9	an	an	DET
ma-287	376	10	integer	integer	NOUN
ma-287	376	11	such	such	DET
ma-287	376	12	that	that	SCONJ
ma-287	376	13	m	m	PROPN
ma-287	376	14	≥	≥	NOUN
ma-287	376	15	1	1	NUM
ma-287	376	16	.	.	PUNCT
ma-287	376	17	figure	figure	NOUN
ma-287	376	18	1	1	NUM
ma-287	376	19	displays	display	VERB
ma-287	376	20	the	the	DET
ma-287	376	21	plots	plot	NOUN
ma-287	376	22	of	of	ADP
ma-287	376	23	ϕ(x	ϕ(x	PROPN
ma-287	376	24	;	;	PUNCT
ma-287	376	25	m	m	X
ma-287	376	26	)	)	PUNCT
ma-287	376	27	for	for	ADP
ma-287	376	28	m	m	PROPN
ma-287	376	29	=	=	SYM
ma-287	376	30	1	1	NUM
ma-287	376	31	,	,	PUNCT
ma-287	376	32	2	2	NUM
ma-287	376	33	,	,	PUNCT
ma-287	376	34	.	.	PUNCT
ma-287	376	35	.	.	PUNCT
ma-287	377	1	.	.	PUNCT
ma-287	378	1	,	,	PUNCT
ma-287	378	2	15	15	NUM
ma-287	378	3	and	and	CCONJ
ma-287	378	4	four	four	NUM
ma-287	378	5	arbitrary	arbitrary	ADJ
ma-287	378	6	values	value	NOUN
ma-287	378	7	of	of	ADP
ma-287	378	8	x	x	X
ma-287	378	9	.	.	PUNCT
ma-287	379	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	379	2	eur	eur	PROPN
ma-287	379	3	.	.	PUNCT
ma-287	380	1	j.	j.	PROPN
ma-287	380	2	math	math	PROPN
ma-287	380	3	.	.	PUNCT
ma-287	381	1	anal	anal	PROPN
ma-287	381	2	.	.	PUNCT
ma-287	382	1	10.28924	10.28924	NUM
ma-287	382	2	/	/	SYM
ma-287	382	3	ada	ada	PROPN
ma-287	382	4	/	/	SYM
ma-287	382	5	ma.5.12	ma.5.12	PROPN
ma-287	382	6	11	11	NUM
ma-287	382	7	0	0	NUM
ma-287	382	8	5	5	NUM
ma-287	382	9	10	10	NUM
ma-287	382	10	15	15	NUM
ma-287	382	11	−	−	NOUN
ma-287	382	12	0	0	NUM
ma-287	382	13	.1	.1	NUM
ma-287	382	14	4	4	NUM
ma-287	382	15	−	−	NOUN
ma-287	382	16	0	0	NUM
ma-287	383	1	.1	.1	NUM
ma-287	383	2	0	0	NUM
ma-287	384	1	−	−	NOUN
ma-287	384	2	0	0	NUM
ma-287	385	1	.0	.0	NUM
ma-287	385	2	6	6	NUM
ma-287	385	3	−	−	NOUN
ma-287	385	4	0	0	NUM
ma-287	386	1	.0	.0	NUM
ma-287	386	2	2	2	NUM
ma-287	386	3	m	m	NOUN
ma-287	386	4	0	0	NUM
ma-287	386	5	5	5	NUM
ma-287	386	6	10	10	NUM
ma-287	386	7	15	15	NUM
ma-287	386	8	0	0	NUM
ma-287	386	9	.0	.0	NUM
ma-287	386	10	0	0	NUM
ma-287	386	11	0	0	NUM
ma-287	387	1	.0	.0	NUM
ma-287	387	2	1	1	NUM
ma-287	387	3	0	0	NUM
ma-287	387	4	.0	.0	NUM
ma-287	387	5	2	2	NUM
ma-287	387	6	0	0	NUM
ma-287	387	7	.0	.0	NUM
ma-287	387	8	3	3	NUM
ma-287	387	9	m	m	NOUN
ma-287	387	10	(	(	PUNCT
ma-287	387	11	a	a	NOUN
ma-287	387	12	)	)	PUNCT
ma-287	387	13	(	(	PUNCT
ma-287	387	14	b	b	NOUN
ma-287	387	15	)	)	PUNCT
ma-287	387	16	0	0	NUM
ma-287	387	17	5	5	NUM
ma-287	387	18	10	10	NUM
ma-287	387	19	15	15	NUM
ma-287	387	20	0	0	NUM
ma-287	387	21	.0	.0	NUM
ma-287	387	22	0	0	NUM
ma-287	388	1	.5	.5	NUM
ma-287	388	2	1	1	NUM
ma-287	388	3	.0	.0	NUM
ma-287	388	4	1	1	NUM
ma-287	388	5	.5	.5	NUM
ma-287	388	6	2	2	NUM
ma-287	388	7	.0	.0	NUM
ma-287	388	8	m	m	NOUN
ma-287	388	9	0	0	NUM
ma-287	388	10	5	5	NUM
ma-287	388	11	10	10	NUM
ma-287	388	12	15	15	NUM
ma-287	388	13	0	0	NUM
ma-287	388	14	5	5	NUM
ma-287	388	15	1	1	NUM
ma-287	388	16	0	0	NUM
ma-287	388	17	1	1	NUM
ma-287	388	18	5	5	NUM
ma-287	388	19	2	2	NUM
ma-287	388	20	0	0	NUM
ma-287	388	21	2	2	NUM
ma-287	388	22	5	5	NUM
ma-287	388	23	3	3	NUM
ma-287	388	24	0	0	NUM
ma-287	388	25	3	3	NUM
ma-287	388	26	5	5	NUM
ma-287	388	27	m	m	NOUN
ma-287	388	28	(	(	PUNCT
ma-287	388	29	c	c	NOUN
ma-287	388	30	)	)	PUNCT
ma-287	388	31	(	(	PUNCT
ma-287	388	32	d	d	X
ma-287	388	33	)	)	PUNCT
ma-287	388	34	figure	figure	NOUN
ma-287	388	35	1	1	NUM
ma-287	388	36	.	.	PUNCT
ma-287	388	37	plots	plot	NOUN
ma-287	388	38	of	of	ADP
ma-287	388	39	ϕ(x	ϕ(x	PROPN
ma-287	388	40	;	;	PUNCT
ma-287	388	41	m	m	X
ma-287	388	42	)	)	PUNCT
ma-287	388	43	for	for	ADP
ma-287	388	44	m	m	PROPN
ma-287	388	45	=	=	SYM
ma-287	388	46	1	1	NUM
ma-287	388	47	,	,	PUNCT
ma-287	388	48	2	2	NUM
ma-287	388	49	,	,	PUNCT
ma-287	388	50	.	.	PUNCT
ma-287	388	51	.	.	PUNCT
ma-287	389	1	.	.	PUNCT
ma-287	390	1	,	,	PUNCT
ma-287	390	2	15	15	NUM
ma-287	390	3	and	and	CCONJ
ma-287	390	4	(	(	PUNCT
ma-287	390	5	a	a	X
ma-287	390	6	)	)	PUNCT
ma-287	390	7	x	x	SYM
ma-287	390	8	=	=	SYM
ma-287	390	9	0.5	0.5	NUM
ma-287	390	10	,	,	PUNCT
ma-287	390	11	(	(	PUNCT
ma-287	390	12	b	b	X
ma-287	390	13	)	)	PUNCT
ma-287	390	14	x	x	SYM
ma-287	391	1	=	=	SYM
ma-287	391	2	1.5	1.5	NUM
ma-287	391	3	,	,	PUNCT
ma-287	391	4	(	(	PUNCT
ma-287	391	5	c	c	X
ma-287	391	6	)	)	PUNCT
ma-287	391	7	x	x	SYM
ma-287	392	1	=	=	SYM
ma-287	392	2	4	4	NUM
ma-287	392	3	,	,	PUNCT
ma-287	392	4	and	and	CCONJ
ma-287	392	5	(	(	PUNCT
ma-287	392	6	d	d	X
ma-287	392	7	)	)	PUNCT
ma-287	392	8	x	x	SYM
ma-287	393	1	=	=	SYM
ma-287	393	2	18	18	NUM
ma-287	393	3	.	.	PUNCT
ma-287	394	1	from	from	ADP
ma-287	394	2	this	this	DET
ma-287	394	3	figure	figure	NOUN
ma-287	394	4	we	we	PRON
ma-287	394	5	see	see	VERB
ma-287	394	6	that	that	SCONJ
ma-287	394	7	,	,	PUNCT
ma-287	394	8	for	for	ADP
ma-287	394	9	all	all	DET
ma-287	394	10	the	the	DET
ma-287	394	11	values	value	NOUN
ma-287	394	12	of	of	ADP
ma-287	394	13	x	x	PUNCT
ma-287	394	14	considered	consider	VERB
ma-287	394	15	,	,	PUNCT
ma-287	394	16	ϕ(x	ϕ(x	PROPN
ma-287	394	17	;	;	PUNCT
ma-287	394	18	m	m	X
ma-287	394	19	)	)	PUNCT
ma-287	394	20	converges	converge	VERB
ma-287	394	21	very	very	ADV
ma-287	394	22	quicklyto	quicklyto	NOUN
ma-287	394	23	0	0	NUM
ma-287	394	24	;	;	PUNCT
ma-287	394	25	it	it	PRON
ma-287	394	26	begins	begin	VERB
ma-287	394	27	to	to	PART
ma-287	394	28	be	be	AUX
ma-287	394	29	very	very	ADV
ma-287	394	30	close	close	ADJ
ma-287	394	31	to	to	ADP
ma-287	394	32	the	the	DET
ma-287	394	33	y	y	NOUN
ma-287	394	34	=	=	SYM
ma-287	394	35	0	0	NUM
ma-287	394	36	axis	axis	NOUN
ma-287	394	37	from	from	ADP
ma-287	394	38	m	m	NOUN
ma-287	394	39	=	=	NOUN
ma-287	394	40	7.let	7.let	SCONJ
ma-287	394	41	us	we	PRON
ma-287	394	42	mention	mention	VERB
ma-287	394	43	that	that	SCONJ
ma-287	394	44	the	the	DET
ma-287	394	45	proof	proof	NOUN
ma-287	394	46	of	of	ADP
ma-287	394	47	the	the	DET
ma-287	394	48	item	item	NOUN
ma-287	394	49	numbered	number	VERB
ma-287	394	50	2	2	NUM
ma-287	394	51	has	have	VERB
ma-287	394	52	a	a	DET
ma-287	394	53	different	different	ADJ
ma-287	394	54	construction	construction	NOUN
ma-287	394	55	than	than	ADP
ma-287	394	56	the	the	DET
ma-287	394	57	onein	onein	NOUN
ma-287	394	58	[	[	X
ma-287	394	59	2	2	NUM
ma-287	394	60	,	,	PUNCT
ma-287	394	61	chapter	chapter	NOUN
ma-287	394	62	31	31	NUM
ma-287	394	63	,	,	PUNCT
ma-287	394	64	page	page	NOUN
ma-287	394	65	399	399	NUM
ma-287	394	66	]	]	PUNCT
ma-287	394	67	,	,	PUNCT
ma-287	394	68	even	even	ADV
ma-287	394	69	though	though	SCONJ
ma-287	394	70	it	it	PRON
ma-287	394	71	is	be	AUX
ma-287	394	72	based	base	VERB
ma-287	394	73	on	on	ADP
ma-287	394	74	the	the	DET
ma-287	394	75	same	same	ADJ
ma-287	394	76	functional	functional	NOUN
ma-287	394	77	basis.also	basis.also	ADV
ma-287	394	78	,	,	PUNCT
ma-287	394	79	from	from	ADP
ma-287	394	80	proposition	proposition	NOUN
ma-287	394	81	2.2	2.2	NUM
ma-287	394	82	,	,	PUNCT
ma-287	394	83	several	several	ADJ
ma-287	394	84	expansions	expansion	NOUN
ma-287	394	85	of	of	ADP
ma-287	394	86	the	the	DET
ma-287	394	87	logarithmic	logarithmic	ADJ
ma-287	394	88	function	function	NOUN
ma-287	394	89	can	can	AUX
ma-287	394	90	be	be	AUX
ma-287	394	91	deduced	deduce	VERB
ma-287	394	92	.	.	PUNCT
ma-287	395	1	forinstance	forinstance	PROPN
ma-287	395	2	,	,	PUNCT
ma-287	395	3	based	base	VERB
ma-287	395	4	on	on	ADP
ma-287	395	5	the	the	DET
ma-287	395	6	item	item	NOUN
ma-287	395	7	numbered	number	VERB
ma-287	395	8	4	4	NUM
ma-287	395	9	,	,	PUNCT
ma-287	395	10	for	for	ADP
ma-287	395	11	x	x	X
ma-287	395	12	≥	≥	NOUN
ma-287	395	13	1	1	NUM
ma-287	395	14	,	,	PUNCT
ma-287	395	15	we	we	PRON
ma-287	395	16	have	have	VERB
ma-287	395	17	log(x	log(x	NUM
ma-287	395	18	)	)	PUNCT
ma-287	396	1	=	=	PUNCT
ma-287	397	1	[	[	PUNCT
ma-287	397	2	√	√	NUM
ma-287	397	3	x	x	PUNCT
ma-287	397	4	−	−	PROPN
ma-287	397	5	1−	1−	NUM
ma-287	398	1	+	+	NOUN
ma-287	398	2	∞∑	∞∑	ADJ
ma-287	398	3	k=1	k=1	X
ma-287	398	4	2(k−1)/2(x2	2(k−1)/2(x2	NOUN
ma-287	398	5	−k	−k	NOUN
ma-287	398	6	−	−	PROPN
ma-287	398	7	1)3/2√	1)3/2√	NUM
ma-287	398	8	1	1	NUM
ma-287	399	1	+	+	CCONJ
ma-287	399	2	x2	x2	NOUN
ma-287	399	3	−k	−k	PROPN
ma-287	399	4	+	+	CCONJ
ma-287	399	5	√	√	NUM
ma-287	399	6	2	2	NUM
ma-287	399	7	]	]	SYM
ma-287	399	8	2	2	NUM
ma-287	399	9	.	.	PUNCT
ma-287	400	1	such	such	ADJ
ma-287	400	2	expansions	expansion	NOUN
ma-287	400	3	are	be	AUX
ma-287	400	4	innovative	innovative	ADJ
ma-287	400	5	,	,	PUNCT
ma-287	400	6	to	to	ADP
ma-287	400	7	the	the	DET
ma-287	400	8	best	good	ADJ
ma-287	400	9	of	of	ADP
ma-287	400	10	our	our	PRON
ma-287	400	11	knowledge.concerning	knowledge.concerne	VERB
ma-287	400	12	the	the	DET
ma-287	400	13	item	item	NOUN
ma-287	400	14	numbered	number	VERB
ma-287	400	15	9	9	NUM
ma-287	400	16	in	in	ADP
ma-287	400	17	proposition	proposition	NOUN
ma-287	400	18	2.2	2.2	NUM
ma-287	400	19	,	,	PUNCT
ma-287	400	20	one	one	PRON
ma-287	400	21	can	can	AUX
ma-287	400	22	remark	remark	VERB
ma-287	400	23	that	that	SCONJ
ma-287	400	24	sinh(x	sinh(x	PROPN
ma-287	400	25	−	−	PROPN
ma-287	400	26	1)−	1)−	PROPN
ma-287	400	27	sinh[log(x	sinh[log(x	PROPN
ma-287	400	28	)	)	PUNCT
ma-287	400	29	]	]	PUNCT
ma-287	401	1	=	=	PUNCT
ma-287	401	2	1	1	NUM
ma-287	401	3	2x	2x	NUM
ma-287	401	4	−	−	NOUN
ma-287	401	5	x	x	SYM
ma-287	401	6	2	2	NUM
ma-287	401	7	−	−	NOUN
ma-287	401	8	sinh(1−	sinh(1−	ADJ
ma-287	401	9	x	x	NOUN
ma-287	401	10	)	)	PUNCT
ma-287	401	11	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	401	12	eur	eur	PROPN
ma-287	401	13	.	.	PUNCT
ma-287	402	1	j.	j.	PROPN
ma-287	402	2	math	math	PROPN
ma-287	402	3	.	.	PUNCT
ma-287	403	1	anal	anal	PROPN
ma-287	403	2	.	.	PUNCT
ma-287	404	1	10.28924	10.28924	NUM
ma-287	404	2	/	/	SYM
ma-287	404	3	ada	ada	PROPN
ma-287	404	4	/	/	SYM
ma-287	404	5	ma.5.12	ma.5.12	PROPN
ma-287	404	6	12and	12and	NOUN
ma-287	404	7	,	,	PUNCT
ma-287	404	8	by	by	ADP
ma-287	404	9	using	use	VERB
ma-287	404	10	the	the	DET
ma-287	404	11	series	series	NOUN
ma-287	404	12	expansion	expansion	NOUN
ma-287	404	13	of	of	ADP
ma-287	404	14	the	the	DET
ma-287	404	15	hyperbolic	hyperbolic	ADJ
ma-287	404	16	sine	sine	NOUN
ma-287	404	17	function	function	NOUN
ma-287	404	18	,	,	PUNCT
ma-287	404	19	the	the	DET
ma-287	404	20	following	follow	VERB
ma-287	404	21	alternative	alternative	ADJ
ma-287	404	22	expres	expre	NOUN
ma-287	404	23	-	-	PUNCT
ma-287	404	24	sion	sion	PROPN
ma-287	404	25	is	be	AUX
ma-287	404	26	obtained	obtain	VERB
ma-287	404	27	:	:	PUNCT
ma-287	404	28	sinh(x	sinh(x	NUM
ma-287	404	29	−	−	PROPN
ma-287	404	30	1)−	1)−	PROPN
ma-287	404	31	sinh[log(x	sinh[log(x	PROPN
ma-287	404	32	)	)	PUNCT
ma-287	404	33	]	]	PUNCT
ma-287	405	1	=	=	PUNCT
ma-287	405	2	1	1	NUM
ma-287	405	3	2x	2x	NUM
ma-287	405	4	−	−	NOUN
ma-287	405	5	x	x	SYM
ma-287	405	6	2	2	NUM
ma-287	405	7	−	−	NOUN
ma-287	405	8	+	+	NOUN
ma-287	405	9	∞∑	∞∑	ADJ
ma-287	405	10	k=0	k=0	PROPN
ma-287	405	11	(	(	PUNCT
ma-287	405	12	1−	1−	NUM
ma-287	405	13	x)2k+1	x)2k+1	NOUN
ma-287	405	14	(	(	PUNCT
ma-287	405	15	2k	2k	NUM
ma-287	405	16	+	+	CCONJ
ma-287	405	17	1	1	NUM
ma-287	405	18	)	)	PUNCT
ma-287	405	19	!	!	PUNCT
ma-287	405	20	.	.	PUNCT
ma-287	406	1	clearly	clearly	ADV
ma-287	406	2	,	,	PUNCT
ma-287	406	3	it	it	PRON
ma-287	406	4	can	can	AUX
ma-287	406	5	be	be	AUX
ma-287	406	6	more	more	ADV
ma-287	406	7	manageable	manageable	ADJ
ma-287	406	8	to	to	PART
ma-287	406	9	use	use	VERB
ma-287	406	10	in	in	ADP
ma-287	406	11	comparison	comparison	NOUN
ma-287	406	12	to	to	ADP
ma-287	406	13	the	the	DET
ma-287	406	14	one	one	NOUN
ma-287	406	15	in	in	ADP
ma-287	406	16	the	the	DET
ma-287	406	17	item	item	NOUN
ma-287	406	18	numbered	number	VERB
ma-287	406	19	9,depending	9,depende	VERB
ma-287	406	20	on	on	ADP
ma-287	406	21	the	the	DET
ma-287	406	22	context	context	NOUN
ma-287	406	23	.	.	PUNCT
ma-287	407	1	the	the	DET
ma-287	407	2	same	same	ADJ
ma-287	407	3	remark	remark	NOUN
ma-287	407	4	holds	hold	VERB
ma-287	407	5	for	for	ADP
ma-287	407	6	the	the	DET
ma-287	407	7	item	item	NOUN
ma-287	407	8	numbered	number	VERB
ma-287	407	9	10	10	NUM
ma-287	407	10	;	;	PUNCT
ma-287	407	11	we	we	PRON
ma-287	407	12	have	have	VERB
ma-287	407	13	cosh(x	cosh(x	NUM
ma-287	407	14	−	−	PROPN
ma-287	407	15	1)−	1)−	PROPN
ma-287	407	16	cosh[log(x	cosh[log(x	NOUN
ma-287	407	17	)	)	PUNCT
ma-287	407	18	]	]	PUNCT
ma-287	408	1	=	=	PUNCT
ma-287	408	2	−	−	PROPN
ma-287	408	3	1	1	NUM
ma-287	408	4	2x	2x	NUM
ma-287	408	5	−	−	NOUN
ma-287	408	6	x	x	SYM
ma-287	408	7	2	2	NUM
ma-287	408	8	+	+	CCONJ
ma-287	408	9	+	+	ADJ
ma-287	408	10	∞∑	∞∑	ADJ
ma-287	408	11	k=0	k=0	PROPN
ma-287	408	12	(	(	PUNCT
ma-287	408	13	1−	1−	NUM
ma-287	408	14	x)2k	x)2k	PRON
ma-287	408	15	(	(	PUNCT
ma-287	408	16	2k	2k	NUM
ma-287	408	17	)	)	PUNCT
ma-287	408	18	!	!	PUNCT
ma-287	408	19	.	.	PUNCT
ma-287	409	1	thus	thus	ADV
ma-287	409	2	,	,	PUNCT
ma-287	409	3	the	the	DET
ma-287	409	4	items	item	NOUN
ma-287	409	5	numbered	number	VERB
ma-287	409	6	9	9	NUM
ma-287	409	7	and	and	CCONJ
ma-287	409	8	10	10	NUM
ma-287	409	9	are	be	AUX
ma-287	409	10	mainly	mainly	ADV
ma-287	409	11	interesting	interesting	ADJ
ma-287	409	12	because	because	SCONJ
ma-287	409	13	of	of	ADP
ma-287	409	14	their	their	PRON
ma-287	409	15	originality	originality	NOUN
ma-287	409	16	,	,	PUNCT
ma-287	409	17	i.e.	i.e.	X
ma-287	409	18	,	,	PUNCT
ma-287	409	19	thetrigonometric	thetrigonometric	NOUN
ma-287	409	20	and	and	CCONJ
ma-287	409	21	hyperbolic	hyperbolic	ADJ
ma-287	409	22	functions	function	NOUN
ma-287	409	23	involved	involve	VERB
ma-287	409	24	,	,	PUNCT
ma-287	409	25	respectively	respectively	ADV
ma-287	409	26	.	.	PUNCT
ma-287	410	1	3	3	X
ma-287	410	2	.	.	X
ma-287	410	3	applications	application	NOUN
ma-287	410	4	some	some	DET
ma-287	410	5	consequences	consequence	NOUN
ma-287	410	6	of	of	ADP
ma-287	410	7	our	our	PRON
ma-287	410	8	results	result	NOUN
ma-287	410	9	are	be	AUX
ma-287	410	10	described	describe	VERB
ma-287	410	11	in	in	ADP
ma-287	410	12	this	this	DET
ma-287	410	13	part	part	NOUN
ma-287	410	14	.	.	PUNCT
ma-287	411	1	3.1	3.1	NUM
ma-287	411	2	.	.	PUNCT
ma-287	411	3	product	product	NOUN
ma-287	411	4	expansions	expansion	NOUN
ma-287	411	5	.	.	PUNCT
ma-287	412	1	proposition	proposition	NOUN
ma-287	412	2	2.2	2.2	NUM
ma-287	412	3	can	can	AUX
ma-287	412	4	be	be	AUX
ma-287	412	5	used	use	VERB
ma-287	412	6	for	for	ADP
ma-287	412	7	many	many	ADJ
ma-287	412	8	purposes	purpose	NOUN
ma-287	412	9	,	,	PUNCT
ma-287	412	10	including	include	VERB
ma-287	412	11	product	product	NOUN
ma-287	412	12	ex	ex	NOUN
ma-287	412	13	-	-	NOUN
ma-287	412	14	pansions	pansion	NOUN
ma-287	412	15	.	.	PUNCT
ma-287	413	1	the	the	DET
ma-287	413	2	result	result	NOUN
ma-287	413	3	below	below	ADV
ma-287	413	4	illustrates	illustrate	VERB
ma-287	413	5	this	this	DET
ma-287	413	6	claim	claim	NOUN
ma-287	413	7	with	with	ADP
ma-287	413	8	an	an	DET
ma-287	413	9	example	example	NOUN
ma-287	413	10	.	.	PUNCT
ma-287	414	1	proposition	proposition	NOUN
ma-287	414	2	3.1	3.1	NUM
ma-287	414	3	.	.	PUNCT
ma-287	415	1	the	the	DET
ma-287	415	2	infinite	infinite	ADJ
ma-287	415	3	product	product	NOUN
ma-287	415	4	expansions	expansion	NOUN
ma-287	415	5	below	below	ADV
ma-287	415	6	are	be	AUX
ma-287	415	7	valid.(1	valid.(1	PROPN
ma-287	415	8	)	)	PUNCT
ma-287	415	9	for	for	ADP
ma-287	415	10	x	x	SYM
ma-287	415	11	>	>	X
ma-287	415	12	1	1	NUM
ma-287	415	13	,	,	PUNCT
ma-287	415	14	we	we	PRON
ma-287	415	15	have	have	VERB
ma-287	415	16	x	x	X
ma-287	415	17	−	−	PROPN
ma-287	415	18	1	1	NUM
ma-287	415	19	log(x	log(x	NOUN
ma-287	415	20	)	)	PUNCT
ma-287	415	21	=	=	PUNCT
ma-287	416	1	+	+	ADJ
ma-287	416	2	∞∏	∞∏	X
ma-287	416	3	k=1	k=1	X
ma-287	416	4	1	1	NUM
ma-287	417	1	+	+	NUM
ma-287	417	2	x2	x2	NOUN
ma-287	417	3	−k	−k	PROPN
ma-287	417	4	2	2	NUM
ma-287	417	5	.	.	PUNCT
ma-287	418	1	this	this	DET
ma-287	418	2	expansion	expansion	NOUN
ma-287	418	3	is	be	AUX
ma-287	418	4	,	,	PUNCT
ma-287	418	5	in	in	ADP
ma-287	418	6	fact	fact	NOUN
ma-287	418	7	,	,	PUNCT
ma-287	418	8	valid	valid	ADJ
ma-287	418	9	for	for	ADP
ma-287	418	10	x	x	PROPN
ma-287	418	11	∈	∈	PROPN
ma-287	418	12	(	(	PUNCT
ma-287	418	13	0,+∞	0,+∞	NUM
ma-287	418	14	)	)	PUNCT
ma-287	418	15	\	\	NOUN
ma-287	418	16	{	{	PUNCT
ma-287	418	17	1	1	NUM
ma-287	418	18	}	}	PUNCT
ma-287	418	19	,	,	PUNCT
ma-287	418	20	as	as	SCONJ
ma-287	418	21	discussed	discuss	VERB
ma-287	418	22	later	later	ADV
ma-287	418	23	.	.	PUNCT
ma-287	419	1	it	it	PRON
ma-287	419	2	"	"	PUNCT
ma-287	419	3	almost	almost	ADV
ma-287	419	4	"	"	PUNCT
ma-287	419	5	corresponds	correspond	VERB
ma-287	419	6	to	to	ADP
ma-287	419	7	the	the	DET
ma-287	419	8	so	so	ADV
ma-287	419	9	-	-	PUNCT
ma-287	419	10	called	call	VERB
ma-287	419	11	seidel	seidel	PROPN
ma-287	419	12	formula	formula	NOUN
ma-287	419	13	(	(	PUNCT
ma-287	419	14	see	see	VERB
ma-287	419	15	[	[	X
ma-287	419	16	15]).(2	15]).(2	NUM
ma-287	419	17	)	)	PUNCT
ma-287	419	18	for	for	ADP
ma-287	419	19	x	x	SYM
ma-287	419	20	>	>	X
ma-287	419	21	1	1	NUM
ma-287	419	22	,	,	PUNCT
ma-287	419	23	we	we	PRON
ma-287	419	24	have	have	VERB
ma-287	419	25	log(x	log(x	NUM
ma-287	419	26	)	)	PUNCT
ma-287	419	27	x	x	SYM
ma-287	420	1	−	−	PROPN
ma-287	420	2	1	1	NUM
ma-287	420	3	=	=	PUNCT
ma-287	421	1	+	+	NOUN
ma-287	421	2	∞∏	∞∏	X
ma-287	421	3	k=1	k=1	X
ma-287	421	4	2	2	NUM
ma-287	421	5	1	1	NUM
ma-287	421	6	+	+	NUM
ma-287	421	7	x2	x2	PROPN
ma-287	421	8	−k	−k	PROPN
ma-287	421	9	.	.	PUNCT
ma-287	422	1	this	this	DET
ma-287	422	2	expansion	expansion	NOUN
ma-287	422	3	is	be	AUX
ma-287	422	4	,	,	PUNCT
ma-287	422	5	in	in	ADP
ma-287	422	6	fact	fact	NOUN
ma-287	422	7	,	,	PUNCT
ma-287	422	8	valid	valid	ADJ
ma-287	422	9	for	for	ADP
ma-287	422	10	x	x	PROPN
ma-287	422	11	∈	∈	PROPN
ma-287	422	12	(	(	PUNCT
ma-287	422	13	0,+∞	0,+∞	NUM
ma-287	422	14	)	)	PUNCT
ma-287	422	15	\	\	NOUN
ma-287	422	16	{	{	PUNCT
ma-287	422	17	1	1	NUM
ma-287	422	18	}	}	PUNCT
ma-287	422	19	,	,	PUNCT
ma-287	422	20	as	as	SCONJ
ma-287	422	21	discussed	discuss	VERB
ma-287	422	22	later.(3	later.(3	NOUN
ma-287	422	23	)	)	PUNCT
ma-287	422	24	for	for	ADP
ma-287	422	25	x	x	SYM
ma-287	422	26	>	>	X
ma-287	422	27	0	0	NUM
ma-287	422	28	,	,	PUNCT
ma-287	422	29	we	we	PRON
ma-287	422	30	have	have	VERB
ma-287	422	31	ex	ex	NOUN
ma-287	423	1	−	−	NOUN
ma-287	423	2	1	1	NUM
ma-287	423	3	x	x	X
ma-287	423	4	=	=	PUNCT
ma-287	424	1	+	+	ADJ
ma-287	424	2	∞∏	∞∏	X
ma-287	424	3	k=1	k=1	X
ma-287	424	4	1	1	NUM
ma-287	424	5	+	+	NUM
ma-287	424	6	e2	e2	NOUN
ma-287	424	7	−kx	−kx	NOUN
ma-287	424	8	2	2	NUM
ma-287	424	9	.	.	PUNCT
ma-287	425	1	this	this	DET
ma-287	425	2	expansion	expansion	NOUN
ma-287	425	3	is	be	AUX
ma-287	425	4	,	,	PUNCT
ma-287	425	5	in	in	ADP
ma-287	425	6	fact	fact	NOUN
ma-287	425	7	,	,	PUNCT
ma-287	425	8	valid	valid	ADJ
ma-287	425	9	for	for	ADP
ma-287	425	10	x	x	PROPN
ma-287	425	11	∈	∈	PROPN
ma-287	425	12	r	r	NOUN
ma-287	425	13	\	\	PUNCT
ma-287	425	14	{	{	PUNCT
ma-287	425	15	0	0	NUM
ma-287	425	16	}	}	PUNCT
ma-287	425	17	,	,	PUNCT
ma-287	425	18	as	as	SCONJ
ma-287	425	19	discussed	discuss	VERB
ma-287	425	20	later.(4	later.(4	PROPN
ma-287	425	21	)	)	PUNCT
ma-287	425	22	for	for	ADP
ma-287	425	23	x	x	SYM
ma-287	425	24	>	>	X
ma-287	425	25	0	0	NUM
ma-287	425	26	,	,	PUNCT
ma-287	425	27	we	we	PRON
ma-287	425	28	have	have	VERB
ma-287	425	29	the	the	DET
ma-287	425	30	following	follow	VERB
ma-287	425	31	product	product	NOUN
ma-287	425	32	expansion	expansion	NOUN
ma-287	425	33	of	of	ADP
ma-287	425	34	the	the	DET
ma-287	425	35	einstein	einstein	PROPN
ma-287	425	36	function	function	NOUN
ma-287	425	37	:	:	PUNCT
ma-287	425	38	e2(x	e2(x	X
ma-287	425	39	)	)	PUNCT
ma-287	425	40	=	=	PUNCT
ma-287	426	1	x	x	X
ma-287	426	2	ex	ex	NOUN
ma-287	426	3	−	−	PROPN
ma-287	426	4	1	1	NUM
ma-287	426	5	=	=	PUNCT
ma-287	427	1	+	+	NOUN
ma-287	427	2	∞∏	∞∏	X
ma-287	427	3	k=1	k=1	X
ma-287	427	4	2	2	NUM
ma-287	427	5	1	1	NUM
ma-287	427	6	+	+	NUM
ma-287	427	7	e2	e2	PROPN
ma-287	427	8	−kx	−kx	NOUN
ma-287	427	9	.	.	PUNCT
ma-287	428	1	this	this	DET
ma-287	428	2	expansion	expansion	NOUN
ma-287	428	3	is	be	AUX
ma-287	428	4	,	,	PUNCT
ma-287	428	5	in	in	ADP
ma-287	428	6	fact	fact	NOUN
ma-287	428	7	,	,	PUNCT
ma-287	428	8	valid	valid	ADJ
ma-287	428	9	for	for	ADP
ma-287	428	10	x	x	PROPN
ma-287	428	11	∈	∈	PROPN
ma-287	428	12	r	r	NOUN
ma-287	428	13	\	\	PUNCT
ma-287	428	14	{	{	PUNCT
ma-287	428	15	0	0	NUM
ma-287	428	16	}	}	PUNCT
ma-287	428	17	,	,	PUNCT
ma-287	428	18	as	as	SCONJ
ma-287	428	19	discussed	discuss	VERB
ma-287	428	20	later	later	ADV
ma-287	428	21	.	.	PUNCT
ma-287	429	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	429	2	eur	eur	PROPN
ma-287	429	3	.	.	PUNCT
ma-287	430	1	j.	j.	PROPN
ma-287	430	2	math	math	PROPN
ma-287	430	3	.	.	PUNCT
ma-287	431	1	anal	anal	PROPN
ma-287	431	2	.	.	PUNCT
ma-287	432	1	10.28924	10.28924	NUM
ma-287	432	2	/	/	SYM
ma-287	432	3	ada	ada	PROPN
ma-287	432	4	/	/	SYM
ma-287	432	5	ma.5.12	ma.5.12	PROPN
ma-287	432	6	13	13	NUM
ma-287	432	7	proof	proof	NOUN
ma-287	432	8	.	.	PUNCT
ma-287	433	1	let	let	VERB
ma-287	433	2	us	we	PRON
ma-287	433	3	prove	prove	VERB
ma-287	433	4	each	each	DET
ma-287	433	5	result	result	NOUN
ma-287	433	6	,	,	PUNCT
ma-287	433	7	one	one	NUM
ma-287	433	8	by	by	ADP
ma-287	433	9	one.(1	one.(1	PROPN
ma-287	433	10	)	)	PUNCT
ma-287	433	11	we	we	PRON
ma-287	433	12	propose	propose	VERB
ma-287	433	13	two	two	NUM
ma-287	433	14	different	different	ADJ
ma-287	433	15	proofs	proof	NOUN
ma-287	433	16	.	.	PUNCT
ma-287	434	1	proof	proof	NOUN
ma-287	434	2	1	1	NUM
ma-287	434	3	:	:	PUNCT
ma-287	434	4	use	use	NOUN
ma-287	434	5	of	of	ADP
ma-287	434	6	proposition	proposition	NOUN
ma-287	434	7	2.2	2.2	NUM
ma-287	434	8	.	.	PUNCT
ma-287	434	9	:	:	PUNCT
ma-287	435	1	for	for	ADP
ma-287	435	2	x	x	SYM
ma-287	435	3	>	>	X
ma-287	435	4	1	1	NUM
ma-287	435	5	,	,	PUNCT
ma-287	435	6	it	it	PRON
ma-287	435	7	follows	follow	VERB
ma-287	435	8	from	from	ADP
ma-287	435	9	the	the	DET
ma-287	435	10	item	item	NOUN
ma-287	435	11	numbered	number	VERB
ma-287	435	12	6	6	NUM
ma-287	435	13	inproposition	inproposition	NOUN
ma-287	435	14	2.2	2.2	NUM
ma-287	436	1	that	that	PRON
ma-287	437	1	log(x	log(x	NUM
ma-287	438	1	−	−	PROPN
ma-287	438	2	1)−	1)−	NUM
ma-287	438	3	log	log	NOUN
ma-287	438	4	[	[	X
ma-287	438	5	log(x	log(x	NOUN
ma-287	438	6	)	)	PUNCT
ma-287	438	7	]	]	PUNCT
ma-287	439	1	=	=	PUNCT
ma-287	440	1	+	+	PUNCT
ma-287	440	2	∞∑	∞∑	NOUN
ma-287	440	3	k=1	k=1	PUNCT
ma-287	440	4	log	log	NOUN
ma-287	440	5	(	(	PUNCT
ma-287	440	6	1	1	NUM
ma-287	440	7	+	+	NUM
ma-287	440	8	x2	x2	NOUN
ma-287	440	9	−k	−k	PROPN
ma-287	440	10	2	2	NUM
ma-287	440	11	)	)	PUNCT
ma-287	440	12	.	.	PUNCT
ma-287	441	1	by	by	ADP
ma-287	441	2	the	the	DET
ma-287	441	3	continuity	continuity	NOUN
ma-287	441	4	of	of	ADP
ma-287	441	5	the	the	DET
ma-287	441	6	logarithmic	logarithmic	ADJ
ma-287	441	7	function	function	NOUN
ma-287	441	8	over	over	ADP
ma-287	441	9	its	its	PRON
ma-287	441	10	domain	domain	NOUN
ma-287	441	11	,	,	PUNCT
ma-287	441	12	it	it	PRON
ma-287	441	13	can	can	AUX
ma-287	441	14	be	be	AUX
ma-287	441	15	rewritten	rewrite	VERB
ma-287	441	16	as	as	ADP
ma-287	441	17	log	log	NOUN
ma-287	441	18	[	[	PUNCT
ma-287	441	19	x	x	SYM
ma-287	441	20	−	−	PROPN
ma-287	441	21	1	1	NUM
ma-287	441	22	log(x	log(x	NOUN
ma-287	441	23	)	)	PUNCT
ma-287	441	24	]	]	PUNCT
ma-287	442	1	=	=	PUNCT
ma-287	442	2	log	log	NOUN
ma-287	442	3	[	[	PUNCT
ma-287	442	4	+	+	NOUN
ma-287	442	5	∞∏	∞∏	X
ma-287	442	6	k=1	k=1	X
ma-287	442	7	1	1	NUM
ma-287	443	1	+	+	NUM
ma-287	443	2	x2	x2	NOUN
ma-287	443	3	−k	−k	PROPN
ma-287	443	4	2	2	NUM
ma-287	443	5	]	]	PUNCT
ma-287	443	6	.	.	PUNCT
ma-287	444	1	composing	compose	VERB
ma-287	444	2	with	with	ADP
ma-287	444	3	the	the	DET
ma-287	444	4	exponential	exponential	ADJ
ma-287	444	5	function	function	NOUN
ma-287	444	6	,	,	PUNCT
ma-287	444	7	we	we	PRON
ma-287	444	8	get	get	VERB
ma-287	444	9	the	the	DET
ma-287	444	10	desired	desire	VERB
ma-287	444	11	result	result	NOUN
ma-287	444	12	,	,	PUNCT
ma-287	444	13	i.e.	i.e.	X
ma-287	444	14	,	,	PUNCT
ma-287	444	15	x	x	SYM
ma-287	444	16	−	−	PROPN
ma-287	444	17	1	1	NUM
ma-287	444	18	log(x	log(x	NOUN
ma-287	444	19	)	)	PUNCT
ma-287	444	20	=	=	PUNCT
ma-287	445	1	+	+	ADJ
ma-287	445	2	∞∏	∞∏	X
ma-287	445	3	k=1	k=1	X
ma-287	445	4	1	1	NUM
ma-287	446	1	+	+	NUM
ma-287	446	2	x2	x2	NOUN
ma-287	446	3	−k	−k	PROPN
ma-287	446	4	2	2	NUM
ma-287	446	5	.	.	PUNCT
ma-287	447	1	proof	proof	NOUN
ma-287	447	2	2	2	NUM
ma-287	447	3	:	:	PUNCT
ma-287	447	4	iterative	iterative	NOUN
ma-287	447	5	scheme	scheme	NOUN
ma-287	447	6	.	.	PUNCT
ma-287	447	7	:	:	PUNCT
ma-287	447	8	to	to	PART
ma-287	447	9	provide	provide	VERB
ma-287	447	10	an	an	DET
ma-287	447	11	alternative	alternative	ADJ
ma-287	447	12	proof	proof	NOUN
ma-287	447	13	,	,	PUNCT
ma-287	447	14	we	we	PRON
ma-287	447	15	now	now	ADV
ma-287	447	16	revisit	revisit	VERB
ma-287	447	17	the	the	DET
ma-287	447	18	originalproof	originalproof	NOUN
ma-287	447	19	of	of	ADP
ma-287	447	20	the	the	DET
ma-287	447	21	seidel	seidel	NOUN
ma-287	447	22	formula	formula	NOUN
ma-287	447	23	in	in	ADP
ma-287	447	24	[	[	X
ma-287	447	25	15	15	NUM
ma-287	447	26	]	]	PUNCT
ma-287	447	27	.	.	PUNCT
ma-287	448	1	we	we	PRON
ma-287	448	2	can	can	AUX
ma-287	448	3	write	write	VERB
ma-287	448	4	x	x	PUNCT
ma-287	448	5	−	−	PROPN
ma-287	448	6	1	1	NUM
ma-287	448	7	=	=	SYM
ma-287	448	8	(	(	PUNCT
ma-287	448	9	1	1	NUM
ma-287	448	10	+	+	CCONJ
ma-287	448	11	√	√	NUM
ma-287	448	12	x	x	NOUN
ma-287	448	13	)	)	PUNCT
ma-287	448	14	(	(	PUNCT
ma-287	448	15	√	√	NUM
ma-287	448	16	x	x	PUNCT
ma-287	448	17	−	−	PROPN
ma-287	448	18	1	1	NUM
ma-287	448	19	)	)	PUNCT
ma-287	448	20	=	=	NOUN
ma-287	449	1	(	(	PUNCT
ma-287	449	2	1	1	NUM
ma-287	449	3	+	+	NUM
ma-287	449	4	x2	x2	ADJ
ma-287	449	5	−1	−1	NOUN
ma-287	449	6	)	)	PUNCT
ma-287	449	7	(	(	PUNCT
ma-287	449	8	x2	x2	NOUN
ma-287	449	9	−1	−1	NOUN
ma-287	449	10	−	−	NOUN
ma-287	449	11	1	1	NUM
ma-287	449	12	)	)	PUNCT
ma-287	449	13	and	and	CCONJ
ma-287	449	14	,	,	PUNCT
ma-287	449	15	with	with	ADP
ma-287	449	16	the	the	DET
ma-287	449	17	same	same	ADJ
ma-287	449	18	principle	principle	NOUN
ma-287	449	19	,	,	PUNCT
ma-287	449	20	we	we	PRON
ma-287	449	21	can	can	AUX
ma-287	449	22	write	write	VERB
ma-287	449	23	the	the	DET
ma-287	449	24	last	last	ADJ
ma-287	449	25	term	term	NOUN
ma-287	449	26	as	as	ADP
ma-287	449	27	x2	x2	PROPN
ma-287	449	28	−1	−1	NOUN
ma-287	449	29	−	−	PROPN
ma-287	449	30	1	1	NUM
ma-287	449	31	=	=	SYM
ma-287	449	32	(	(	PUNCT
ma-287	449	33	1	1	NUM
ma-287	449	34	+	+	CCONJ
ma-287	449	35	x2−2)(x2−2	x2−2)(x2−2	NUM
ma-287	450	1	−	−	PROPN
ma-287	450	2	1	1	NUM
ma-287	450	3	)	)	PUNCT
ma-287	450	4	,	,	PUNCT
ma-287	450	5	and	and	CCONJ
ma-287	450	6	the	the	DET
ma-287	450	7	same	same	ADJ
ma-287	450	8	for	for	ADP
ma-287	450	9	x2−2	x2−2	PROPN
ma-287	450	10	−	−	PROPN
ma-287	450	11	1	1	NUM
ma-287	450	12	,	,	PUNCT
ma-287	450	13	etc	etc	X
ma-287	450	14	.	.	X
ma-287	451	1	so	so	ADV
ma-287	451	2	,	,	PUNCT
ma-287	451	3	for	for	ADP
ma-287	451	4	any	any	DET
ma-287	451	5	integer	integer	NOUN
ma-287	451	6	n	n	PRON
ma-287	451	7	≥	≥	NOUN
ma-287	451	8	1	1	NUM
ma-287	451	9	,	,	PUNCT
ma-287	451	10	we	we	PRON
ma-287	451	11	have	have	VERB
ma-287	451	12	x	x	X
ma-287	451	13	−	−	PROPN
ma-287	451	14	1	1	NUM
ma-287	451	15	=	=	SYM
ma-287	451	16	(	(	PUNCT
ma-287	451	17	1	1	NUM
ma-287	452	1	+	+	NUM
ma-287	452	2	x2−1)(x2−1	x2−1)(x2−1	NUM
ma-287	452	3	−	−	NOUN
ma-287	452	4	1	1	NUM
ma-287	452	5	)	)	PUNCT
ma-287	452	6	=	=	NOUN
ma-287	452	7	(	(	PUNCT
ma-287	452	8	1	1	NUM
ma-287	452	9	+	+	CCONJ
ma-287	452	10	x2−1)(1	x2−1)(1	PRON
ma-287	453	1	+	+	CCONJ
ma-287	453	2	x2−2)(x2−2	x2−2)(x2−2	NUM
ma-287	454	1	−	−	NUM
ma-287	454	2	1	1	NUM
ma-287	454	3	)	)	PUNCT
ma-287	454	4	=	=	PUNCT
ma-287	454	5	.	.	PUNCT
ma-287	454	6	.	.	PUNCT
ma-287	454	7	.	.	PUNCT
ma-287	455	1	=	=	PUNCT
ma-287	456	1	[	[	PUNCT
ma-287	456	2	n∏	n∏	PROPN
ma-287	456	3	k=1	k=1	PROPN
ma-287	456	4	(	(	PUNCT
ma-287	456	5	1	1	NUM
ma-287	456	6	+	+	NUM
ma-287	456	7	x2	x2	NOUN
ma-287	456	8	−k	−k	PROPN
ma-287	456	9	)	)	PUNCT
ma-287	456	10	]	]	PUNCT
ma-287	457	1	(	(	PUNCT
ma-287	457	2	x2	x2	NOUN
ma-287	457	3	−n	−n	NOUN
ma-287	457	4	−	−	PROPN
ma-287	457	5	1	1	NUM
ma-287	457	6	)	)	PUNCT
ma-287	457	7	=	=	NOUN
ma-287	458	1	[	[	PUNCT
ma-287	458	2	n∏	n∏	NOUN
ma-287	458	3	k=1	k=1	NOUN
ma-287	458	4	1	1	NUM
ma-287	459	1	+	+	NUM
ma-287	459	2	x2	x2	NOUN
ma-287	459	3	−k	−k	PROPN
ma-287	459	4	2	2	NUM
ma-287	459	5	]	]	PUNCT
ma-287	459	6	2n(x2	2n(x2	NUM
ma-287	459	7	−n	−n	NOUN
ma-287	459	8	−	−	PROPN
ma-287	459	9	1	1	NUM
ma-287	459	10	)	)	PUNCT
ma-287	459	11	.	.	PUNCT
ma-287	460	1	therefore	therefore	ADV
ma-287	460	2	,	,	PUNCT
ma-287	460	3	by	by	SCONJ
ma-287	460	4	considering	consider	VERB
ma-287	460	5	the	the	DET
ma-287	460	6	limit	limit	NOUN
ma-287	460	7	when	when	SCONJ
ma-287	460	8	n	n	X
ma-287	460	9	→	→	SYM
ma-287	460	10	+	+	PROPN
ma-287	460	11	∞	∞	PROPN
ma-287	460	12	,	,	PUNCT
ma-287	460	13	we	we	PRON
ma-287	460	14	obtain	obtain	VERB
ma-287	460	15	x	x	PUNCT
ma-287	460	16	−	−	PROPN
ma-287	460	17	1	1	NUM
ma-287	460	18	=	=	SYM
ma-287	460	19	[	[	PUNCT
ma-287	460	20	lim	lim	PROPN
ma-287	460	21	n→+∞	n→+∞	PROPN
ma-287	460	22	n∏	n∏	PROPN
ma-287	460	23	k=1	k=1	NOUN
ma-287	460	24	1	1	NUM
ma-287	461	1	+	+	NUM
ma-287	461	2	x2	x2	NOUN
ma-287	461	3	−k	−k	PROPN
ma-287	461	4	2	2	NUM
ma-287	461	5	]	]	PUNCT
ma-287	461	6	[	[	PUNCT
ma-287	461	7	lim	lim	PROPN
ma-287	461	8	n→+∞	n→+∞	VERB
ma-287	461	9	2n(x2	2n(x2	NUM
ma-287	461	10	−n	−n	NOUN
ma-287	461	11	−	−	PROPN
ma-287	461	12	1	1	NUM
ma-287	461	13	)	)	PUNCT
ma-287	461	14	]	]	PUNCT
ma-287	462	1	=	=	PUNCT
ma-287	463	1	[	[	PUNCT
ma-287	463	2	+	+	NOUN
ma-287	463	3	∞∏	∞∏	X
ma-287	463	4	k=1	k=1	X
ma-287	463	5	1	1	NUM
ma-287	463	6	+	+	NUM
ma-287	463	7	x2	x2	NOUN
ma-287	463	8	−k	−k	PROPN
ma-287	463	9	2	2	NUM
ma-287	463	10	]	]	PUNCT
ma-287	463	11	log(x	log(x	NUM
ma-287	463	12	)	)	PUNCT
ma-287	463	13	,	,	PUNCT
ma-287	463	14	which	which	PRON
ma-287	463	15	implies	imply	VERB
ma-287	463	16	that	that	SCONJ
ma-287	463	17	x	x	PUNCT
ma-287	463	18	−	−	NOUN
ma-287	463	19	1	1	NUM
ma-287	463	20	log(x	log(x	NOUN
ma-287	463	21	)	)	PUNCT
ma-287	463	22	=	=	PUNCT
ma-287	464	1	+	+	ADJ
ma-287	464	2	∞∏	∞∏	X
ma-287	464	3	k=1	k=1	X
ma-287	464	4	1	1	NUM
ma-287	465	1	+	+	NUM
ma-287	465	2	x2	x2	NOUN
ma-287	465	3	−k	−k	PROPN
ma-287	465	4	2	2	NUM
ma-287	465	5	.	.	PUNCT
ma-287	466	1	thus	thus	ADV
ma-287	466	2	,	,	PUNCT
ma-287	466	3	proof	proof	NOUN
ma-287	466	4	1	1	NUM
ma-287	466	5	offers	offer	VERB
ma-287	466	6	an	an	DET
ma-287	466	7	alternative	alternative	NOUN
ma-287	466	8	to	to	ADP
ma-287	466	9	these	these	DET
ma-287	466	10	known	know	VERB
ma-287	466	11	developments	development	NOUN
ma-287	466	12	by	by	ADP
ma-287	466	13	using	use	VERB
ma-287	466	14	proposition2.2	proposition2.2	NOUN
ma-287	466	15	.	.	PUNCT
ma-287	467	1	proof	proof	NOUN
ma-287	467	2	2	2	NUM
ma-287	467	3	,	,	PUNCT
ma-287	467	4	however	however	ADV
ma-287	467	5	,	,	PUNCT
ma-287	467	6	has	have	VERB
ma-287	467	7	the	the	DET
ma-287	467	8	advantage	advantage	NOUN
ma-287	467	9	of	of	ADP
ma-287	467	10	being	be	AUX
ma-287	467	11	valid	valid	ADJ
ma-287	467	12	for	for	ADP
ma-287	467	13	x	x	PROPN
ma-287	467	14	∈	∈	PROPN
ma-287	467	15	(	(	PUNCT
ma-287	467	16	0,+∞	0,+∞	NUM
ma-287	467	17	)	)	PUNCT
ma-287	467	18	\	\	NOUN
ma-287	467	19	{	{	PUNCT
ma-287	467	20	1	1	NUM
ma-287	467	21	}	}	PUNCT
ma-287	467	22	,	,	PUNCT
ma-287	468	1	not	not	PART
ma-287	468	2	just	just	ADV
ma-287	468	3	x	x	SYM
ma-287	468	4	>	>	X
ma-287	468	5	1	1	X
ma-287	468	6	.	.	PUNCT
ma-287	468	7	we	we	PRON
ma-287	468	8	will	will	AUX
ma-287	468	9	show	show	VERB
ma-287	468	10	later	later	ADV
ma-287	468	11	why	why	SCONJ
ma-287	468	12	this	this	PRON
ma-287	468	13	is	be	AUX
ma-287	468	14	actually	actually	ADV
ma-287	468	15	not	not	PART
ma-287	468	16	a	a	DET
ma-287	468	17	problem.(2	problem.(2	NOUN
ma-287	468	18	)	)	PUNCT
ma-287	468	19	for	for	ADP
ma-287	468	20	x	x	SYM
ma-287	468	21	>	>	X
ma-287	468	22	1	1	NUM
ma-287	468	23	,	,	PUNCT
ma-287	468	24	by	by	ADP
ma-287	468	25	using	use	VERB
ma-287	468	26	the	the	DET
ma-287	468	27	previous	previous	ADJ
ma-287	468	28	result	result	NOUN
ma-287	468	29	,	,	PUNCT
ma-287	468	30	we	we	PRON
ma-287	468	31	get	get	VERB
ma-287	468	32	log(x	log(x	PROPN
ma-287	468	33	)	)	PUNCT
ma-287	468	34	x	x	SYM
ma-287	469	1	−	−	PROPN
ma-287	469	2	1	1	NUM
ma-287	469	3	=	=	SYM
ma-287	469	4	1	1	NUM
ma-287	469	5	(	(	PUNCT
ma-287	469	6	x	x	SYM
ma-287	469	7	−	−	PROPN
ma-287	469	8	1)/	1)/	NUM
ma-287	469	9	log(x	log(x	PROPN
ma-287	469	10	)	)	PUNCT
ma-287	469	11	=	=	PUNCT
ma-287	470	1	1∏+∞	1∏+∞	NUM
ma-287	470	2	k=1	k=1	X
ma-287	470	3	[	[	PUNCT
ma-287	470	4	(	(	PUNCT
ma-287	470	5	1	1	NUM
ma-287	470	6	+	+	NUM
ma-287	470	7	x2	x2	NOUN
ma-287	470	8	−k	−k	PROPN
ma-287	470	9	)	)	PUNCT
ma-287	470	10	/2	/2	PUNCT
ma-287	470	11	]	]	PUNCT
ma-287	471	1	=	=	PUNCT
ma-287	472	1	+	+	ADJ
ma-287	472	2	∞∏	∞∏	X
ma-287	472	3	k=1	k=1	X
ma-287	472	4	1	1	NUM
ma-287	472	5	(	(	PUNCT
ma-287	472	6	1	1	NUM
ma-287	472	7	+	+	NUM
ma-287	472	8	x2	x2	NOUN
ma-287	472	9	−k	−k	ADJ
ma-287	472	10	)	)	PUNCT
ma-287	472	11	/2	/2	PUNCT
ma-287	473	1	=	=	PUNCT
ma-287	474	1	+	+	ADJ
ma-287	474	2	∞∏	∞∏	X
ma-287	474	3	k=1	k=1	X
ma-287	474	4	2	2	NUM
ma-287	474	5	1	1	NUM
ma-287	474	6	+	+	NUM
ma-287	474	7	x2	x2	PROPN
ma-287	474	8	−k	−k	PROPN
ma-287	474	9	.	.	PUNCT
ma-287	475	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	475	2	eur	eur	PROPN
ma-287	475	3	.	.	PUNCT
ma-287	476	1	j.	j.	PROPN
ma-287	476	2	math	math	PROPN
ma-287	476	3	.	.	PUNCT
ma-287	477	1	anal	anal	PROPN
ma-287	477	2	.	.	PUNCT
ma-287	478	1	10.28924	10.28924	NUM
ma-287	478	2	/	/	SYM
ma-287	478	3	ada	ada	PROPN
ma-287	478	4	/	/	SYM
ma-287	478	5	ma.5.12	ma.5.12	PROPN
ma-287	478	6	14(3	14(3	NUM
ma-287	478	7	)	)	PUNCT
ma-287	478	8	for	for	ADP
ma-287	478	9	y	y	PROPN
ma-287	478	10	>	>	X
ma-287	478	11	0	0	PROPN
ma-287	478	12	,	,	PUNCT
ma-287	478	13	by	by	ADP
ma-287	478	14	applying	apply	VERB
ma-287	478	15	the	the	DET
ma-287	478	16	result	result	NOUN
ma-287	478	17	in	in	ADP
ma-287	478	18	the	the	DET
ma-287	478	19	item	item	NOUN
ma-287	478	20	numbered	number	VERB
ma-287	478	21	1	1	NUM
ma-287	478	22	with	with	ADP
ma-287	478	23	x	x	PROPN
ma-287	478	24	=	=	SYM
ma-287	478	25	ey	ey	PROPN
ma-287	478	26	>	>	X
ma-287	478	27	1	1	NUM
ma-287	478	28	,	,	PUNCT
ma-287	478	29	we	we	PRON
ma-287	478	30	establishthat	establishthat	VERB
ma-287	478	31	ey	ey	INTJ
ma-287	478	32	−	−	PROPN
ma-287	479	1	1	1	NUM
ma-287	479	2	y	y	NOUN
ma-287	479	3	=	=	PUNCT
ma-287	479	4	x	x	SYM
ma-287	479	5	−	−	PROPN
ma-287	479	6	1	1	NUM
ma-287	479	7	log(x	log(x	NOUN
ma-287	479	8	)	)	PUNCT
ma-287	479	9	=	=	PUNCT
ma-287	480	1	+	+	ADJ
ma-287	480	2	∞∏	∞∏	X
ma-287	480	3	k=1	k=1	X
ma-287	480	4	1	1	NUM
ma-287	481	1	+	+	NUM
ma-287	481	2	x2	x2	NOUN
ma-287	481	3	−k	−k	ADJ
ma-287	481	4	2	2	NUM
ma-287	481	5	=	=	SYM
ma-287	481	6	+	+	NOUN
ma-287	481	7	∞∏	∞∏	X
ma-287	481	8	k=1	k=1	X
ma-287	481	9	1	1	NUM
ma-287	481	10	+	+	NUM
ma-287	481	11	e2	e2	NOUN
ma-287	481	12	−ky	−ky	NOUN
ma-287	481	13	2	2	NUM
ma-287	481	14	.	.	PUNCT
ma-287	482	1	(	(	PUNCT
ma-287	482	2	4	4	NUM
ma-287	482	3	)	)	PUNCT
ma-287	482	4	for	for	ADP
ma-287	482	5	x	x	SYM
ma-287	482	6	>	>	X
ma-287	482	7	0	0	NUM
ma-287	482	8	,	,	PUNCT
ma-287	482	9	by	by	ADP
ma-287	482	10	using	use	VERB
ma-287	482	11	the	the	DET
ma-287	482	12	result	result	NOUN
ma-287	482	13	in	in	ADP
ma-287	482	14	the	the	DET
ma-287	482	15	previous	previous	ADJ
ma-287	482	16	item	item	NOUN
ma-287	482	17	,	,	PUNCT
ma-287	482	18	we	we	PRON
ma-287	482	19	have	have	VERB
ma-287	482	20	e2(x	e2(x	NOUN
ma-287	482	21	)	)	PUNCT
ma-287	482	22	=	=	PUNCT
ma-287	483	1	x	x	X
ma-287	483	2	ex	ex	NOUN
ma-287	483	3	−	−	PROPN
ma-287	483	4	1	1	NUM
ma-287	483	5	=	=	SYM
ma-287	483	6	1	1	NUM
ma-287	483	7	[	[	X
ma-287	483	8	(	(	PUNCT
ma-287	483	9	ex	ex	ADJ
ma-287	483	10	−	−	PROPN
ma-287	483	11	1)/x	1)/x	NUM
ma-287	483	12	]	]	PUNCT
ma-287	483	13	=	=	PUNCT
ma-287	483	14	1∏+∞	1∏+∞	NUM
ma-287	484	1	k=1[(1	k=1[(1	INTJ
ma-287	485	1	+	+	CCONJ
ma-287	485	2	e	e	X
ma-287	485	3	2−kx)/2	2−kx)/2	NUM
ma-287	485	4	]	]	X
ma-287	485	5	=	=	PUNCT
ma-287	486	1	+	+	ADJ
ma-287	486	2	∞∏	∞∏	X
ma-287	486	3	k=1	k=1	X
ma-287	486	4	1	1	NUM
ma-287	486	5	(	(	PUNCT
ma-287	486	6	1	1	NUM
ma-287	486	7	+	+	NUM
ma-287	486	8	e2	e2	PROPN
ma-287	486	9	−kx)/2	−kx)/2	X
ma-287	487	1	=	=	PUNCT
ma-287	488	1	+	+	NOUN
ma-287	488	2	∞∏	∞∏	X
ma-287	488	3	k=1	k=1	X
ma-287	488	4	2	2	NUM
ma-287	488	5	1	1	NUM
ma-287	488	6	+	+	NUM
ma-287	488	7	e2	e2	PROPN
ma-287	488	8	−kx	−kx	NOUN
ma-287	488	9	.	.	PUNCT
ma-287	489	1	this	this	PRON
ma-287	489	2	ends	end	VERB
ma-287	489	3	the	the	DET
ma-287	489	4	proof	proof	NOUN
ma-287	489	5	.	.	PUNCT
ma-287	490	1	�	�	PROPN
ma-287	490	2	from	from	ADP
ma-287	490	3	the	the	DET
ma-287	490	4	item	item	NOUN
ma-287	490	5	numbered	number	VERB
ma-287	490	6	2	2	NUM
ma-287	490	7	,	,	PUNCT
ma-287	490	8	for	for	ADP
ma-287	490	9	x	x	X
ma-287	490	10	≥	≥	NOUN
ma-287	490	11	1	1	NUM
ma-287	490	12	,	,	PUNCT
ma-287	490	13	we	we	PRON
ma-287	490	14	get	get	VERB
ma-287	490	15	the	the	DET
ma-287	490	16	following	follow	VERB
ma-287	490	17	product	product	NOUN
ma-287	490	18	expansion	expansion	NOUN
ma-287	490	19	of	of	ADP
ma-287	490	20	the	the	DET
ma-287	490	21	logarithmicfunction	logarithmicfunction	NOUN
ma-287	490	22	:	:	PUNCT
ma-287	490	23	log(x	log(x	X
ma-287	490	24	)	)	PUNCT
ma-287	490	25	=	=	SYM
ma-287	491	1	(	(	PUNCT
ma-287	491	2	x	x	SYM
ma-287	491	3	−	−	NOUN
ma-287	491	4	1	1	NUM
ma-287	491	5	)	)	PUNCT
ma-287	492	1	+	+	NOUN
ma-287	492	2	∞∏	∞∏	X
ma-287	492	3	k=1	k=1	X
ma-287	492	4	2	2	NUM
ma-287	492	5	1	1	NUM
ma-287	492	6	+	+	CCONJ
ma-287	492	7	x2	x2	NOUN
ma-287	492	8	−k	−k	PROPN
ma-287	492	9	,	,	PUNCT
ma-287	492	10	(	(	PUNCT
ma-287	492	11	3	3	X
ma-287	492	12	)	)	PUNCT
ma-287	492	13	which	which	PRON
ma-287	492	14	corresponds	correspond	VERB
ma-287	492	15	to	to	ADP
ma-287	492	16	the	the	DET
ma-287	492	17	seidel	seidel	PROPN
ma-287	492	18	formula	formula	NOUN
ma-287	492	19	restricted	restrict	VERB
ma-287	492	20	to	to	ADP
ma-287	492	21	(	(	PUNCT
ma-287	492	22	1,+∞	1,+∞	NUM
ma-287	492	23	)	)	PUNCT
ma-287	492	24	.	.	PUNCT
ma-287	493	1	we	we	PRON
ma-287	493	2	can	can	AUX
ma-287	493	3	complete	complete	VERB
ma-287	493	4	it	it	PRON
ma-287	493	5	for	for	ADP
ma-287	493	6	x	x	PROPN
ma-287	493	7	∈	∈	PROPN
ma-287	493	8	(	(	PUNCT
ma-287	493	9	0	0	NUM
ma-287	493	10	,	,	PUNCT
ma-287	493	11	1)based	1)based	NUM
ma-287	493	12	on	on	ADP
ma-287	493	13	the	the	DET
ma-287	493	14	case	case	NOUN
ma-287	493	15	x	x	X
ma-287	493	16	≥	≥	NUM
ma-287	493	17	1	1	NUM
ma-287	493	18	.	.	PUNCT
ma-287	494	1	indeed	indeed	ADV
ma-287	494	2	,	,	PUNCT
ma-287	494	3	for	for	ADP
ma-287	494	4	x	x	PROPN
ma-287	494	5	∈	∈	PROPN
ma-287	494	6	(	(	PUNCT
ma-287	494	7	0	0	NUM
ma-287	494	8	,	,	PUNCT
ma-287	494	9	1	1	NUM
ma-287	494	10	)	)	PUNCT
ma-287	494	11	,	,	PUNCT
ma-287	494	12	since	since	SCONJ
ma-287	494	13	1	1	NUM
ma-287	494	14	/	/	SYM
ma-287	494	15	x	x	SYM
ma-287	494	16	>	>	X
ma-287	494	17	1	1	NUM
ma-287	494	18	and	and	CCONJ
ma-287	494	19	∑+∞k=1	∑+∞k=1	PROPN
ma-287	494	20	2−k	2−k	NUM
ma-287	495	1	=	=	SYM
ma-287	495	2	1	1	NUM
ma-287	495	3	,	,	PUNCT
ma-287	495	4	we	we	PRON
ma-287	495	5	get	get	VERB
ma-287	495	6	log(x	log(x	PRON
ma-287	495	7	)	)	PUNCT
ma-287	495	8	=	=	SYM
ma-287	496	1	−	−	PROPN
ma-287	496	2	log	log	NOUN
ma-287	496	3	(	(	PUNCT
ma-287	496	4	1	1	NUM
ma-287	496	5	x	x	NOUN
ma-287	496	6	)	)	PUNCT
ma-287	496	7	=	=	SYM
ma-287	496	8	−	−	PROPN
ma-287	496	9	(	(	PUNCT
ma-287	496	10	1	1	NUM
ma-287	496	11	x	x	SYM
ma-287	496	12	−	−	NOUN
ma-287	496	13	1	1	NUM
ma-287	496	14	)	)	PUNCT
ma-287	497	1	+	+	VERB
ma-287	497	2	∞∏	∞∏	X
ma-287	497	3	k=1	k=1	X
ma-287	497	4	2	2	NUM
ma-287	497	5	1	1	NUM
ma-287	497	6	+	+	CCONJ
ma-287	497	7	x−2−k	x−2−k	PUNCT
ma-287	497	8	=	=	PRON
ma-287	498	1	(	(	PUNCT
ma-287	498	2	x	x	SYM
ma-287	498	3	−	−	PROPN
ma-287	498	4	1	1	NUM
ma-287	498	5	)	)	PUNCT
ma-287	498	6	1	1	NUM
ma-287	498	7	x	x	SYM
ma-287	498	8	[	[	PUNCT
ma-287	498	9	+	+	NOUN
ma-287	498	10	∞∏	∞∏	X
ma-287	498	11	k=1	k=1	X
ma-287	498	12	2	2	NUM
ma-287	498	13	1	1	NUM
ma-287	498	14	+	+	CCONJ
ma-287	498	15	x2	x2	NOUN
ma-287	498	16	−k	−k	PROPN
ma-287	498	17	]	]	X
ma-287	498	18	[	[	PUNCT
ma-287	498	19	+	+	NOUN
ma-287	498	20	∞∏	∞∏	X
ma-287	498	21	k=1	k=1	PUNCT
ma-287	499	1	x2	x2	PROPN
ma-287	499	2	−k	−k	VERB
ma-287	499	3	]	]	PUNCT
ma-287	500	1	=	=	PUNCT
ma-287	500	2	(	(	PUNCT
ma-287	500	3	x	x	SYM
ma-287	500	4	−	−	PROPN
ma-287	500	5	1	1	NUM
ma-287	500	6	)	)	PUNCT
ma-287	500	7	[	[	PUNCT
ma-287	500	8	+	+	NOUN
ma-287	500	9	∞∏	∞∏	X
ma-287	500	10	k=1	k=1	X
ma-287	500	11	2x2	2x2	NUM
ma-287	500	12	−k	−k	VERB
ma-287	500	13	1	1	NUM
ma-287	500	14	+	+	CCONJ
ma-287	500	15	x2	x2	NOUN
ma-287	500	16	−k	−k	NOUN
ma-287	500	17	]	]	PUNCT
ma-287	500	18	[	[	PUNCT
ma-287	500	19	x−1	x−1	NOUN
ma-287	500	20	+	+	NOUN
ma-287	500	21	∑+∞	∑+∞	ADJ
ma-287	500	22	k=1	k=1	ADP
ma-287	500	23	2	2	NUM
ma-287	500	24	−k	−k	NOUN
ma-287	500	25	]	]	PUNCT
ma-287	501	1	=	=	PUNCT
ma-287	502	1	(	(	PUNCT
ma-287	502	2	x	x	SYM
ma-287	502	3	−	−	NOUN
ma-287	502	4	1	1	NUM
ma-287	502	5	)	)	PUNCT
ma-287	503	1	+	+	NOUN
ma-287	503	2	∞∏	∞∏	X
ma-287	503	3	k=1	k=1	X
ma-287	503	4	2	2	NUM
ma-287	503	5	1	1	NUM
ma-287	503	6	+	+	NUM
ma-287	503	7	x2	x2	PROPN
ma-287	503	8	−k	−k	ADV
ma-287	503	9	.	.	PUNCT
ma-287	504	1	we	we	PRON
ma-287	504	2	thus	thus	ADV
ma-287	504	3	find	find	VERB
ma-287	504	4	the	the	DET
ma-287	504	5	seidel	seidel	NOUN
ma-287	504	6	formula	formula	NOUN
ma-287	504	7	in	in	ADP
ma-287	504	8	its	its	PRON
ma-287	504	9	entirety	entirety	NOUN
ma-287	504	10	,	,	PUNCT
ma-287	504	11	as	as	SCONJ
ma-287	504	12	mentioned	mention	VERB
ma-287	504	13	in	in	ADP
ma-287	504	14	the	the	DET
ma-287	504	15	second	second	ADJ
ma-287	504	16	proof	proof	NOUN
ma-287	504	17	of	of	ADP
ma-287	504	18	the	the	DET
ma-287	504	19	itemnumbered	itemnumbere	VERB
ma-287	504	20	1	1	NUM
ma-287	504	21	in	in	ADP
ma-287	504	22	proposition	proposition	NOUN
ma-287	504	23	3.1	3.1	NUM
ma-287	504	24	.	.	PUNCT
ma-287	505	1	the	the	DET
ma-287	505	2	advantages	advantage	NOUN
ma-287	505	3	of	of	ADP
ma-287	505	4	this	this	DET
ma-287	505	5	decomposition	decomposition	NOUN
ma-287	505	6	are	be	AUX
ma-287	505	7	that	that	SCONJ
ma-287	505	8	it	it	PRON
ma-287	505	9	has	have	VERB
ma-287	505	10	no	no	DET
ma-287	505	11	constraintson	constraintson	NOUN
ma-287	505	12	the	the	DET
ma-287	505	13	natural	natural	ADJ
ma-287	505	14	domain	domain	NOUN
ma-287	505	15	of	of	ADP
ma-287	505	16	definition	definition	NOUN
ma-287	505	17	,	,	PUNCT
ma-287	505	18	i.e.	i.e.	X
ma-287	505	19	,	,	PUNCT
ma-287	505	20	x	x	X
ma-287	505	21	>	>	X
ma-287	505	22	0	0	NUM
ma-287	505	23	,	,	PUNCT
ma-287	505	24	to	to	PART
ma-287	505	25	satisfy	satisfy	VERB
ma-287	505	26	log(x	log(x	PROPN
ma-287	505	27	)	)	PUNCT
ma-287	505	28	=	=	SYM
ma-287	506	1	−	−	NOUN
ma-287	506	2	log(1	log(1	NOUN
ma-287	506	3	/	/	SYM
ma-287	506	4	x	x	NOUN
ma-287	506	5	)	)	PUNCT
ma-287	506	6	,	,	PUNCT
ma-287	506	7	which	which	PRON
ma-287	506	8	is	be	AUX
ma-287	506	9	not	not	PART
ma-287	506	10	the	the	DET
ma-287	506	11	casefor	casefor	NOUN
ma-287	506	12	most	most	ADJ
ma-287	506	13	series	series	NOUN
ma-287	506	14	expansions	expansion	NOUN
ma-287	506	15	of	of	ADP
ma-287	506	16	log(x	log(x	NOUN
ma-287	506	17	)	)	PUNCT
ma-287	506	18	,	,	PUNCT
ma-287	506	19	and	and	CCONJ
ma-287	506	20	also	also	ADV
ma-287	506	21	that	that	DET
ma-287	506	22	log(x	log(x	X
ma-287	506	23	)	)	PUNCT
ma-287	506	24	and	and	CCONJ
ma-287	506	25	x	x	SYM
ma-287	506	26	−	−	NOUN
ma-287	506	27	1	1	NUM
ma-287	506	28	have	have	VERB
ma-287	506	29	the	the	DET
ma-287	506	30	same	same	ADJ
ma-287	506	31	sign	sign	NOUN
ma-287	506	32	accordingto	accordingto	NOUN
ma-287	506	33	x	x	X
ma-287	506	34	∈	∈	PROPN
ma-287	506	35	(	(	PUNCT
ma-287	506	36	0	0	NUM
ma-287	506	37	,	,	PUNCT
ma-287	506	38	1	1	NUM
ma-287	506	39	)	)	PUNCT
ma-287	506	40	and	and	CCONJ
ma-287	506	41	x	x	X
ma-287	506	42	>	>	X
ma-287	506	43	1	1	X
ma-287	506	44	.	.	PUNCT
ma-287	507	1	it	it	PRON
ma-287	507	2	is	be	AUX
ma-287	507	3	also	also	ADV
ma-287	507	4	underexploited	underexploite	VERB
ma-287	507	5	in	in	ADP
ma-287	507	6	the	the	DET
ma-287	507	7	literature	literature	NOUN
ma-287	507	8	to	to	PART
ma-287	507	9	determine	determine	VERB
ma-287	507	10	sharp	sharp	ADJ
ma-287	507	11	logarithmicinequalities	logarithmicinequalitie	NOUN
ma-287	507	12	.	.	PUNCT
ma-287	508	1	we	we	PRON
ma-287	508	2	will	will	AUX
ma-287	508	3	emphasize	emphasize	VERB
ma-287	508	4	this	this	DET
ma-287	508	5	aspect	aspect	NOUN
ma-287	508	6	in	in	ADP
ma-287	508	7	the	the	DET
ma-287	508	8	next	next	ADJ
ma-287	508	9	section	section	NOUN
ma-287	508	10	.	.	PUNCT
ma-287	509	1	thus	thus	ADV
ma-287	509	2	,	,	PUNCT
ma-287	509	3	in	in	ADP
ma-287	509	4	a	a	DET
ma-287	509	5	sense	sense	NOUN
ma-287	509	6	,	,	PUNCT
ma-287	509	7	the	the	DET
ma-287	509	8	results	result	NOUN
ma-287	509	9	intheorem	intheorem	VERB
ma-287	509	10	2.1	2.1	NUM
ma-287	509	11	unify	unify	VERB
ma-287	509	12	three	three	NUM
ma-287	509	13	known	know	VERB
ma-287	509	14	results	result	NOUN
ma-287	509	15	,	,	PUNCT
ma-287	509	16	one	one	NUM
ma-287	509	17	by	by	ADP
ma-287	509	18	srinivasa	srinivasa	PROPN
ma-287	509	19	ramanujan	ramanujan	PROPN
ma-287	509	20	in	in	ADP
ma-287	509	21	[	[	X
ma-287	509	22	12	12	NUM
ma-287	509	23	]	]	PUNCT
ma-287	509	24	,	,	PUNCT
ma-287	509	25	one	one	NUM
ma-287	509	26	by	by	ADP
ma-287	509	27	ludwig	ludwig	PROPN
ma-287	509	28	seidelin	seidelin	PROPN
ma-287	510	1	[	[	X
ma-287	510	2	15	15	NUM
ma-287	510	3	]	]	PUNCT
ma-287	510	4	,	,	PUNCT
ma-287	510	5	and	and	CCONJ
ma-287	510	6	a	a	DET
ma-287	510	7	more	more	ADV
ma-287	510	8	recent	recent	ADJ
ma-287	510	9	one	one	NUM
ma-287	510	10	by	by	ADP
ma-287	510	11	david	david	PROPN
ma-287	510	12	m.	m.	PROPN
ma-287	510	13	bradley	bradley	PROPN
ma-287	510	14	in	in	ADP
ma-287	510	15	[	[	X
ma-287	510	16	4].as	4].as	PRON
ma-287	510	17	an	an	DET
ma-287	510	18	additional	additional	ADJ
ma-287	510	19	numerical	numerical	ADJ
ma-287	510	20	contribution	contribution	NOUN
ma-287	510	21	,	,	PUNCT
ma-287	510	22	let	let	VERB
ma-287	510	23	us	we	PRON
ma-287	510	24	illustrate	illustrate	VERB
ma-287	510	25	this	this	DET
ma-287	510	26	expansion	expansion	NOUN
ma-287	510	27	by	by	ADP
ma-287	510	28	considering	consider	VERB
ma-287	510	29	thefollowing	thefollowe	VERB
ma-287	510	30	function	function	NOUN
ma-287	510	31	:	:	PUNCT
ma-287	510	32	ζ(x	ζ(x	NOUN
ma-287	510	33	;	;	PUNCT
ma-287	510	34	m	m	X
ma-287	510	35	)	)	PUNCT
ma-287	511	1	=	=	SYM
ma-287	511	2	log(x)−	log(x)−	PROPN
ma-287	511	3	(	(	PUNCT
ma-287	511	4	x	x	SYM
ma-287	511	5	−	−	PROPN
ma-287	511	6	1	1	NUM
ma-287	511	7	)	)	PUNCT
ma-287	511	8	m∏	m∏	PROPN
ma-287	511	9	k=1	k=1	NOUN
ma-287	511	10	2	2	NUM
ma-287	511	11	1	1	NUM
ma-287	511	12	+	+	CCONJ
ma-287	511	13	x2	x2	PROPN
ma-287	511	14	−k	−k	PROPN
ma-287	511	15	,	,	PUNCT
ma-287	511	16	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	511	17	eur	eur	PROPN
ma-287	511	18	.	.	PUNCT
ma-287	512	1	j.	j.	PROPN
ma-287	512	2	math	math	PROPN
ma-287	512	3	.	.	PUNCT
ma-287	513	1	anal	anal	PROPN
ma-287	513	2	.	.	PUNCT
ma-287	514	1	10.28924	10.28924	NUM
ma-287	514	2	/	/	SYM
ma-287	514	3	ada	ada	PROPN
ma-287	514	4	/	/	SYM
ma-287	514	5	ma.5.12	ma.5.12	PROPN
ma-287	515	1	15where	15where	NUM
ma-287	515	2	m	m	VERB
ma-287	515	3	denotes	denote	VERB
ma-287	515	4	an	an	DET
ma-287	515	5	integer	integer	NOUN
ma-287	515	6	such	such	DET
ma-287	515	7	that	that	SCONJ
ma-287	515	8	m	m	PROPN
ma-287	515	9	≥	≥	NOUN
ma-287	515	10	1	1	NUM
ma-287	515	11	.	.	PUNCT
ma-287	515	12	figure	figure	NOUN
ma-287	515	13	2	2	NUM
ma-287	515	14	displays	display	VERB
ma-287	515	15	the	the	DET
ma-287	515	16	plots	plot	NOUN
ma-287	515	17	of	of	ADP
ma-287	515	18	ζ(x	ζ(x	PROPN
ma-287	515	19	;	;	PUNCT
ma-287	515	20	m	m	X
ma-287	515	21	)	)	PUNCT
ma-287	515	22	for	for	ADP
ma-287	515	23	m	m	PROPN
ma-287	515	24	=	=	SYM
ma-287	515	25	1	1	NUM
ma-287	515	26	,	,	PUNCT
ma-287	515	27	2	2	NUM
ma-287	515	28	,	,	PUNCT
ma-287	515	29	.	.	PUNCT
ma-287	515	30	.	.	PUNCT
ma-287	516	1	.	.	PUNCT
ma-287	517	1	,	,	PUNCT
ma-287	517	2	15	15	NUM
ma-287	517	3	and	and	CCONJ
ma-287	517	4	four	four	NUM
ma-287	517	5	arbitrary	arbitrary	ADJ
ma-287	517	6	values	value	NOUN
ma-287	517	7	of	of	ADP
ma-287	517	8	x	x	PRON
ma-287	517	9	,	,	PUNCT
ma-287	517	10	including	include	VERB
ma-287	517	11	x	x	X
ma-287	517	12	=	=	SYM
ma-287	517	13	0.5	0.5	NUM
ma-287	517	14	∈	∈	NOUN
ma-287	517	15	(	(	PUNCT
ma-287	517	16	0	0	NUM
ma-287	517	17	,	,	PUNCT
ma-287	517	18	1	1	NUM
ma-287	517	19	)	)	PUNCT
ma-287	517	20	to	to	PART
ma-287	517	21	check	check	VERB
ma-287	517	22	the	the	DET
ma-287	517	23	previousstatement	previousstatement	NOUN
ma-287	517	24	.	.	PUNCT
ma-287	518	1	0	0	NUM
ma-287	518	2	5	5	NUM
ma-287	518	3	10	10	NUM
ma-287	518	4	15	15	NUM
ma-287	518	5	−	−	NOUN
ma-287	518	6	0	0	NUM
ma-287	518	7	.1	.1	NUM
ma-287	518	8	0	0	NUM
ma-287	519	1	−	−	NOUN
ma-287	519	2	0	0	NUM
ma-287	520	1	.0	.0	NUM
ma-287	520	2	6	6	NUM
ma-287	520	3	−	−	NOUN
ma-287	520	4	0	0	NUM
ma-287	521	1	.0	.0	NUM
ma-287	521	2	2	2	NUM
ma-287	521	3	m	m	NOUN
ma-287	521	4	0	0	NUM
ma-287	521	5	5	5	NUM
ma-287	521	6	10	10	NUM
ma-287	521	7	15	15	NUM
ma-287	521	8	−	−	NOUN
ma-287	521	9	0	0	NUM
ma-287	521	10	.0	.0	NUM
ma-287	521	11	4	4	NUM
ma-287	521	12	−	−	NOUN
ma-287	521	13	0	0	NUM
ma-287	522	1	.0	.0	NUM
ma-287	522	2	3	3	NUM
ma-287	522	3	−	−	NOUN
ma-287	522	4	0	0	NUM
ma-287	523	1	.0	.0	NUM
ma-287	523	2	2	2	NUM
ma-287	523	3	−	−	NOUN
ma-287	523	4	0	0	NUM
ma-287	523	5	.0	.0	NUM
ma-287	524	1	1	1	NUM
ma-287	524	2	0	0	NUM
ma-287	524	3	.0	.0	NUM
ma-287	524	4	0	0	NUM
ma-287	524	5	m	m	VERB
ma-287	524	6	(	(	PUNCT
ma-287	524	7	a	a	NOUN
ma-287	524	8	)	)	PUNCT
ma-287	524	9	(	(	PUNCT
ma-287	524	10	b	b	NOUN
ma-287	524	11	)	)	PUNCT
ma-287	524	12	0	0	NUM
ma-287	524	13	5	5	NUM
ma-287	524	14	10	10	NUM
ma-287	524	15	15	15	NUM
ma-287	524	16	−	−	NOUN
ma-287	524	17	0	0	NUM
ma-287	525	1	.6	.6	NUM
ma-287	526	1	−	−	NOUN
ma-287	526	2	0	0	NUM
ma-287	527	1	.4	.4	NUM
ma-287	528	1	−	−	NOUN
ma-287	528	2	0	0	NUM
ma-287	529	1	.2	.2	NUM
ma-287	529	2	0	0	NUM
ma-287	530	1	.0	.0	NUM
ma-287	530	2	m	m	NOUN
ma-287	530	3	0	0	NUM
ma-287	530	4	5	5	NUM
ma-287	530	5	10	10	NUM
ma-287	530	6	15	15	NUM
ma-287	530	7	−	−	NUM
ma-287	530	8	3	3	NUM
ma-287	530	9	.5	.5	NUM
ma-287	531	1	−	−	NUM
ma-287	531	2	2	2	NUM
ma-287	531	3	.5	.5	NUM
ma-287	532	1	−	−	NOUN
ma-287	532	2	1	1	NUM
ma-287	532	3	.5	.5	NUM
ma-287	533	1	−	−	NOUN
ma-287	533	2	0	0	NUM
ma-287	534	1	.5	.5	NUM
ma-287	534	2	m	m	VERB
ma-287	534	3	(	(	PUNCT
ma-287	534	4	c	c	NOUN
ma-287	534	5	)	)	PUNCT
ma-287	534	6	(	(	PUNCT
ma-287	534	7	d	d	X
ma-287	534	8	)	)	PUNCT
ma-287	534	9	figure	figure	NOUN
ma-287	534	10	2	2	NUM
ma-287	534	11	.	.	PUNCT
ma-287	534	12	plots	plot	NOUN
ma-287	534	13	of	of	ADP
ma-287	534	14	ζ(x	ζ(x	PROPN
ma-287	534	15	;	;	PUNCT
ma-287	534	16	m	m	X
ma-287	534	17	)	)	PUNCT
ma-287	534	18	for	for	ADP
ma-287	534	19	m	m	PROPN
ma-287	534	20	=	=	SYM
ma-287	534	21	1	1	NUM
ma-287	534	22	,	,	PUNCT
ma-287	534	23	2	2	NUM
ma-287	534	24	,	,	PUNCT
ma-287	534	25	.	.	PUNCT
ma-287	534	26	.	.	PUNCT
ma-287	535	1	.	.	PUNCT
ma-287	536	1	,	,	PUNCT
ma-287	536	2	15	15	NUM
ma-287	536	3	and	and	CCONJ
ma-287	536	4	(	(	PUNCT
ma-287	536	5	a	a	X
ma-287	536	6	)	)	PUNCT
ma-287	536	7	x	x	SYM
ma-287	536	8	=	=	SYM
ma-287	536	9	0.5	0.5	NUM
ma-287	536	10	,	,	PUNCT
ma-287	536	11	(	(	PUNCT
ma-287	536	12	b	b	X
ma-287	536	13	)	)	PUNCT
ma-287	536	14	x	x	SYM
ma-287	537	1	=	=	SYM
ma-287	537	2	1.5	1.5	NUM
ma-287	537	3	,	,	PUNCT
ma-287	537	4	(	(	PUNCT
ma-287	537	5	c	c	X
ma-287	537	6	)	)	PUNCT
ma-287	537	7	x	x	SYM
ma-287	538	1	=	=	SYM
ma-287	538	2	4	4	NUM
ma-287	538	3	,	,	PUNCT
ma-287	538	4	and	and	CCONJ
ma-287	538	5	(	(	PUNCT
ma-287	538	6	d	d	X
ma-287	538	7	)	)	PUNCT
ma-287	538	8	x	x	SYM
ma-287	538	9	=	=	NOUN
ma-287	538	10	18	18	NUM
ma-287	538	11	.	.	PUNCT
ma-287	539	1	this	this	DET
ma-287	539	2	figure	figure	NOUN
ma-287	539	3	shows	show	VERB
ma-287	539	4	that	that	SCONJ
ma-287	539	5	ζ(x	ζ(x	NOUN
ma-287	539	6	;	;	PUNCT
ma-287	539	7	m	m	X
ma-287	539	8	)	)	PUNCT
ma-287	539	9	converges	converge	VERB
ma-287	539	10	very	very	ADV
ma-287	539	11	quickly	quickly	ADV
ma-287	539	12	to	to	ADP
ma-287	539	13	0	0	NUM
ma-287	539	14	for	for	ADP
ma-287	539	15	all	all	DET
ma-287	539	16	the	the	DET
ma-287	539	17	considered	considered	ADJ
ma-287	539	18	values	value	NOUN
ma-287	539	19	of	of	ADP
ma-287	539	20	x	x	SYM
ma-287	539	21	;	;	PUNCT
ma-287	539	22	itstarts	itstart	NOUN
ma-287	539	23	very	very	ADV
ma-287	539	24	close	close	ADV
ma-287	539	25	to	to	ADP
ma-287	539	26	the	the	DET
ma-287	539	27	axis	axis	NOUN
ma-287	539	28	y	y	PROPN
ma-287	539	29	=	=	NOUN
ma-287	539	30	0	0	NUM
ma-287	539	31	from	from	ADP
ma-287	539	32	m	m	NOUN
ma-287	539	33	=	=	SYM
ma-287	539	34	7.sophisticated	7.sophisticated	NUM
ma-287	539	35	infinite	infinite	ADJ
ma-287	539	36	product	product	NOUN
ma-287	539	37	formulas	formula	NOUN
ma-287	539	38	can	can	AUX
ma-287	539	39	be	be	AUX
ma-287	539	40	derived	derive	VERB
ma-287	539	41	from	from	ADP
ma-287	539	42	the	the	DET
ma-287	539	43	natural	natural	ADJ
ma-287	539	44	properties	property	NOUN
ma-287	539	45	of	of	ADP
ma-287	539	46	the	the	DET
ma-287	539	47	loga	loga	NOUN
ma-287	539	48	-	-	PUNCT
ma-287	539	49	rithmic	rithmic	ADJ
ma-287	539	50	function	function	NOUN
ma-287	539	51	and	and	CCONJ
ma-287	539	52	equation	equation	NOUN
ma-287	539	53	(	(	PUNCT
ma-287	539	54	3	3	NUM
ma-287	539	55	)	)	PUNCT
ma-287	539	56	.	.	PUNCT
ma-287	540	1	in	in	ADP
ma-287	540	2	particular	particular	ADJ
ma-287	540	3	,	,	PUNCT
ma-287	540	4	the	the	DET
ma-287	540	5	formulas	formula	NOUN
ma-287	540	6	below	below	ADV
ma-287	540	7	are	be	AUX
ma-287	540	8	true	true	ADJ
ma-287	540	9	.	.	PUNCT
ma-287	541	1	•	•	NUM
ma-287	541	2	for	for	ADP
ma-287	541	3	x	x	PUNCT
ma-287	541	4	>	>	X
ma-287	541	5	0	0	NUM
ma-287	541	6	,	,	PUNCT
ma-287	541	7	we	we	PRON
ma-287	541	8	have	have	VERB
ma-287	541	9	x	x	X
ma-287	541	10	=	=	SYM
ma-287	541	11	e	e	X
ma-287	541	12	log(x	log(x	PROPN
ma-287	541	13	)	)	PUNCT
ma-287	541	14	,	,	PUNCT
ma-287	541	15	which	which	PRON
ma-287	541	16	is	be	AUX
ma-287	541	17	equivalent	equivalent	ADJ
ma-287	541	18	to	to	ADP
ma-287	541	19	x	x	PUNCT
ma-287	541	20	=	=	SYM
ma-287	541	21	e	e	X
ma-287	541	22	(	(	PUNCT
ma-287	541	23	x−1	x−1	NOUN
ma-287	541	24	)	)	PUNCT
ma-287	541	25	∏+∞	∏+∞	NOUN
ma-287	541	26	k=1	k=1	ADP
ma-287	541	27	2	2	NUM
ma-287	541	28	1+x2	1+x2	NUM
ma-287	541	29	−k	−k	ADV
ma-287	541	30	.	.	PUNCT
ma-287	542	1	•	•	NOUN
ma-287	542	2	for	for	ADP
ma-287	542	3	x	x	PUNCT
ma-287	542	4	>	>	X
ma-287	542	5	0	0	PUNCT
ma-287	542	6	and	and	CCONJ
ma-287	542	7	y	y	PROPN
ma-287	542	8	∈	∈	PROPN
ma-287	542	9	r	r	NOUN
ma-287	542	10	,	,	PUNCT
ma-287	542	11	we	we	PRON
ma-287	542	12	have	have	VERB
ma-287	542	13	log(xy	log(xy	NOUN
ma-287	542	14	)	)	PUNCT
ma-287	543	1	=	=	PUNCT
ma-287	543	2	y	y	PROPN
ma-287	543	3	log(x	log(x	PROPN
ma-287	543	4	)	)	PUNCT
ma-287	543	5	,	,	PUNCT
ma-287	543	6	which	which	PRON
ma-287	543	7	yields	yield	VERB
ma-287	543	8	(	(	PUNCT
ma-287	543	9	xy	xy	NOUN
ma-287	543	10	−	−	PROPN
ma-287	543	11	1	1	X
ma-287	543	12	)	)	PUNCT
ma-287	543	13	+	+	NOUN
ma-287	543	14	∞∏	∞∏	X
ma-287	543	15	k=1	k=1	X
ma-287	543	16	2	2	NUM
ma-287	543	17	1	1	NUM
ma-287	543	18	+	+	NUM
ma-287	543	19	x2	x2	ADJ
ma-287	543	20	−ky	−ky	NOUN
ma-287	543	21	=	=	PUNCT
ma-287	543	22	y(x	y(x	NOUN
ma-287	544	1	−	−	NOUN
ma-287	544	2	1	1	X
ma-287	544	3	)	)	PUNCT
ma-287	545	1	+	+	NOUN
ma-287	545	2	∞∏	∞∏	X
ma-287	545	3	k=1	k=1	X
ma-287	545	4	2	2	NUM
ma-287	545	5	1	1	NUM
ma-287	545	6	+	+	NUM
ma-287	545	7	x2	x2	PROPN
ma-287	545	8	−k	−k	PROPN
ma-287	545	9	.	.	PUNCT
ma-287	546	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	546	2	eur	eur	PROPN
ma-287	546	3	.	.	PUNCT
ma-287	547	1	j.	j.	PROPN
ma-287	547	2	math	math	PROPN
ma-287	547	3	.	.	PUNCT
ma-287	548	1	anal	anal	PROPN
ma-287	548	2	.	.	PUNCT
ma-287	549	1	10.28924	10.28924	NUM
ma-287	549	2	/	/	SYM
ma-287	549	3	ada	ada	PROPN
ma-287	549	4	/	/	SYM
ma-287	549	5	ma.5.12	ma.5.12	PROPN
ma-287	549	6	16similarly	16similarly	NUM
ma-287	549	7	,	,	PUNCT
ma-287	549	8	for	for	ADP
ma-287	549	9	x	x	X
ma-287	549	10	>	>	X
ma-287	549	11	0	0	PUNCT
ma-287	549	12	and	and	CCONJ
ma-287	549	13	y	y	PROPN
ma-287	549	14	∈	∈	PROPN
ma-287	549	15	r	r	NOUN
ma-287	549	16	\	\	PUNCT
ma-287	549	17	{	{	PUNCT
ma-287	549	18	0	0	NUM
ma-287	549	19	}	}	PUNCT
ma-287	549	20	,	,	PUNCT
ma-287	549	21	we	we	PRON
ma-287	549	22	have	have	VERB
ma-287	549	23	log(x	log(x	NUM
ma-287	549	24	)	)	PUNCT
ma-287	549	25	=	=	SYM
ma-287	550	1	(	(	PUNCT
ma-287	550	2	1	1	NUM
ma-287	550	3	/	/	SYM
ma-287	550	4	y	y	NOUN
ma-287	550	5	)	)	PUNCT
ma-287	550	6	log(xy	log(xy	NOUN
ma-287	550	7	)	)	PUNCT
ma-287	550	8	,	,	PUNCT
ma-287	550	9	giving	give	VERB
ma-287	550	10	the	the	DET
ma-287	550	11	followingexpansions	followingexpansion	NOUN
ma-287	550	12	:	:	PUNCT
ma-287	550	13	log(x	log(x	X
ma-287	550	14	)	)	PUNCT
ma-287	550	15	=	=	SYM
ma-287	550	16	1	1	NUM
ma-287	550	17	y	y	NOUN
ma-287	550	18	(	(	PUNCT
ma-287	550	19	xy	xy	PROPN
ma-287	550	20	−	−	PROPN
ma-287	550	21	1	1	X
ma-287	550	22	)	)	PUNCT
ma-287	550	23	+	+	NOUN
ma-287	550	24	∞∏	∞∏	X
ma-287	550	25	k=1	k=1	X
ma-287	550	26	2	2	NUM
ma-287	550	27	1	1	NUM
ma-287	550	28	+	+	CCONJ
ma-287	550	29	x2	x2	ADJ
ma-287	550	30	−ky	−ky	NOUN
ma-287	550	31	.	.	PUNCT
ma-287	551	1	in	in	ADP
ma-287	551	2	particular	particular	ADJ
ma-287	551	3	,	,	PUNCT
ma-287	551	4	for	for	ADP
ma-287	551	5	y	y	PROPN
ma-287	551	6	=	=	SYM
ma-287	551	7	2	2	NUM
ma-287	551	8	m	m	VERB
ma-287	551	9	with	with	ADP
ma-287	551	10	an	an	DET
ma-287	551	11	arbitrary	arbitrary	ADJ
ma-287	551	12	m	m	NOUN
ma-287	551	13	>	>	X
ma-287	551	14	0	0	NUM
ma-287	551	15	,	,	PUNCT
ma-287	551	16	we	we	PRON
ma-287	551	17	have	have	VERB
ma-287	551	18	log(x	log(x	NUM
ma-287	551	19	)	)	PUNCT
ma-287	551	20	=	=	SYM
ma-287	552	1	1	1	NUM
ma-287	552	2	2	2	NUM
ma-287	552	3	m	m	VERB
ma-287	552	4	(	(	PUNCT
ma-287	552	5	x2	x2	NOUN
ma-287	552	6	m	m	VERB
ma-287	552	7	−	−	NOUN
ma-287	552	8	1	1	NUM
ma-287	552	9	)	)	PUNCT
ma-287	553	1	+	+	NOUN
ma-287	553	2	∞∏	∞∏	X
ma-287	553	3	k=1	k=1	X
ma-287	553	4	2	2	NUM
ma-287	553	5	1	1	NUM
ma-287	553	6	+	+	NUM
ma-287	553	7	x2	x2	PROPN
ma-287	553	8	m−k	m−k	NOUN
ma-287	553	9	.	.	PUNCT
ma-287	554	1	•	•	X
ma-287	554	2	for	for	ADP
ma-287	554	3	x	x	PUNCT
ma-287	554	4	>	>	X
ma-287	554	5	0	0	PUNCT
ma-287	555	1	and	and	CCONJ
ma-287	555	2	y	y	PROPN
ma-287	555	3	>	>	X
ma-287	555	4	0	0	PROPN
ma-287	555	5	,	,	PUNCT
ma-287	555	6	we	we	PRON
ma-287	555	7	have	have	VERB
ma-287	555	8	log(xy	log(xy	NOUN
ma-287	555	9	)	)	PUNCT
ma-287	556	1	=	=	SYM
ma-287	556	2	log(x	log(x	X
ma-287	556	3	)	)	PUNCT
ma-287	556	4	+	+	NUM
ma-287	556	5	log(y	log(y	PROPN
ma-287	556	6	)	)	PUNCT
ma-287	556	7	,	,	PUNCT
ma-287	556	8	which	which	PRON
ma-287	556	9	is	be	AUX
ma-287	556	10	equivalent	equivalent	ADJ
ma-287	556	11	to	to	ADP
ma-287	556	12	(	(	PUNCT
ma-287	556	13	xy	xy	NOUN
ma-287	556	14	−	−	PROPN
ma-287	556	15	1	1	X
ma-287	556	16	)	)	PUNCT
ma-287	557	1	+	+	NOUN
ma-287	557	2	∞∏	∞∏	X
ma-287	557	3	k=1	k=1	X
ma-287	557	4	2	2	NUM
ma-287	557	5	1	1	NUM
ma-287	557	6	+	+	CCONJ
ma-287	557	7	(	(	PUNCT
ma-287	557	8	xy)2	xy)2	NOUN
ma-287	557	9	−k	−k	PROPN
ma-287	557	10	=	=	PUNCT
ma-287	558	1	(	(	PUNCT
ma-287	558	2	x	x	SYM
ma-287	558	3	−	−	NOUN
ma-287	558	4	1	1	NUM
ma-287	558	5	)	)	PUNCT
ma-287	559	1	+	+	NOUN
ma-287	559	2	∞∏	∞∏	X
ma-287	559	3	k=1	k=1	X
ma-287	559	4	2	2	NUM
ma-287	559	5	1	1	NUM
ma-287	559	6	+	+	CCONJ
ma-287	559	7	x2	x2	NOUN
ma-287	559	8	−k	−k	PROPN
ma-287	559	9	+	+	CCONJ
ma-287	559	10	(	(	PUNCT
ma-287	559	11	y	y	PROPN
ma-287	559	12	−	−	PROPN
ma-287	559	13	1	1	NUM
ma-287	559	14	)	)	PUNCT
ma-287	560	1	+	+	NOUN
ma-287	560	2	∞∏	∞∏	X
ma-287	560	3	k=1	k=1	X
ma-287	560	4	2	2	NUM
ma-287	560	5	1	1	NUM
ma-287	560	6	+	+	CCONJ
ma-287	560	7	y2	y2	NOUN
ma-287	560	8	−k	−k	ADJ
ma-287	560	9	.	.	PUNCT
ma-287	561	1	in	in	ADP
ma-287	561	2	addition	addition	NOUN
ma-287	561	3	,	,	PUNCT
ma-287	561	4	it	it	PRON
ma-287	561	5	is	be	AUX
ma-287	561	6	known	know	VERB
ma-287	561	7	that	that	SCONJ
ma-287	561	8	the	the	DET
ma-287	561	9	einstein	einstein	ADJ
ma-287	561	10	function	function	NOUN
ma-287	561	11	can	can	AUX
ma-287	561	12	be	be	AUX
ma-287	561	13	expressed	express	VERB
ma-287	561	14	as	as	ADP
ma-287	561	15	a	a	DET
ma-287	561	16	series	series	NOUN
ma-287	561	17	expansion	expansion	NOUN
ma-287	561	18	involvingbernoulli	involvingbernoulli	ADJ
ma-287	561	19	numbers	number	NOUN
ma-287	561	20	as	as	ADP
ma-287	561	21	e2(x	e2(x	NOUN
ma-287	561	22	)	)	PUNCT
ma-287	561	23	=	=	PUNCT
ma-287	562	1	+	+	ADP
ma-287	562	2	∞∑	∞∑	PRON
ma-287	562	3	k=0	k=0	PROPN
ma-287	562	4	b−k	b−k	PROPN
ma-287	562	5	k	k	PROPN
ma-287	562	6	!	!	PUNCT
ma-287	562	7	xk	xk	PROPN
ma-287	562	8	,	,	PUNCT
ma-287	563	1	where	where	SCONJ
ma-287	563	2	b−k	b−k	PROPN
ma-287	563	3	=	=	SYM
ma-287	563	4	k∑	k∑	NOUN
ma-287	563	5	`	`	PUNCT
ma-287	563	6	=	=	SYM
ma-287	563	7	0	0	NUM
ma-287	563	8	∑̀	∑̀	NOUN
ma-287	563	9	v=0	v=0	NOUN
ma-287	563	10	(	(	PUNCT
ma-287	563	11	−1)v	−1)v	X
ma-287	563	12	(	(	PUNCT
ma-287	563	13	`	`	PUNCT
ma-287	563	14	v	v	NOUN
ma-287	563	15	)	)	PUNCT
ma-287	563	16	v	v	NOUN
ma-287	564	1	k	k	PROPN
ma-287	564	2	`	`	PUNCT
ma-287	564	3	+	+	X
ma-287	564	4	1	1	NUM
ma-287	564	5	and	and	CCONJ
ma-287	564	6	(	(	PUNCT
ma-287	564	7	`	`	PUNCT
ma-287	564	8	v	v	NOUN
ma-287	564	9	)	)	PUNCT
ma-287	564	10	=	=	PUNCT
ma-287	564	11	`	`	PUNCT
ma-287	564	12	!	!	PUNCT
ma-287	564	13	/[v	/[v	PUNCT
ma-287	564	14	!	!	PUNCT
ma-287	565	1	(	(	PUNCT
ma-287	565	2	`	`	PUNCT
ma-287	565	3	−	−	PRON
ma-287	565	4	v)!].in	v)!].in	PROPN
ma-287	565	5	some	some	DET
ma-287	565	6	sense	sense	NOUN
ma-287	565	7	,	,	PUNCT
ma-287	565	8	item	item	NOUN
ma-287	565	9	numbered	number	VERB
ma-287	565	10	4	4	NUM
ma-287	565	11	completes	complete	VERB
ma-287	565	12	this	this	DET
ma-287	565	13	result	result	NOUN
ma-287	565	14	by	by	ADP
ma-287	565	15	investigating	investigate	VERB
ma-287	565	16	a	a	DET
ma-287	565	17	simple	simple	ADJ
ma-287	565	18	product	product	NOUN
ma-287	565	19	expansionfor	expansionfor	ADP
ma-287	565	20	x	x	PUNCT
ma-287	565	21	>	>	X
ma-287	565	22	0	0	NUM
ma-287	565	23	,	,	PUNCT
ma-287	565	24	given	give	VERB
ma-287	565	25	as	as	ADP
ma-287	565	26	e2(x	e2(x	NOUN
ma-287	565	27	)	)	PUNCT
ma-287	565	28	=	=	PUNCT
ma-287	566	1	+	+	ADJ
ma-287	566	2	∞∏	∞∏	X
ma-287	566	3	k=1	k=1	X
ma-287	566	4	2	2	NUM
ma-287	566	5	1	1	NUM
ma-287	566	6	+	+	NUM
ma-287	566	7	e2	e2	PROPN
ma-287	566	8	−kx	−kx	NOUN
ma-287	566	9	.	.	PUNCT
ma-287	567	1	(	(	PUNCT
ma-287	567	2	4	4	X
ma-287	567	3	)	)	PUNCT
ma-287	567	4	it	it	PRON
ma-287	567	5	is	be	AUX
ma-287	567	6	interesting	interesting	ADJ
ma-287	567	7	to	to	PART
ma-287	567	8	note	note	VERB
ma-287	567	9	that	that	SCONJ
ma-287	567	10	this	this	DET
ma-287	567	11	formula	formula	NOUN
ma-287	567	12	is	be	AUX
ma-287	567	13	also	also	ADV
ma-287	567	14	valid	valid	ADJ
ma-287	567	15	for	for	ADP
ma-287	567	16	x	x	PUNCT
ma-287	567	17	<	<	X
ma-287	567	18	0	0	NUM
ma-287	567	19	.	.	PUNCT
ma-287	568	1	indeed	indeed	ADV
ma-287	568	2	,	,	PUNCT
ma-287	568	3	in	in	ADP
ma-287	568	4	this	this	DET
ma-287	568	5	case	case	NOUN
ma-287	568	6	,	,	PUNCT
ma-287	568	7	we	we	PRON
ma-287	568	8	can	can	AUX
ma-287	568	9	remarkthat	remarkthat	VERB
ma-287	568	10	e2(x	e2(x	NOUN
ma-287	568	11	)	)	PUNCT
ma-287	568	12	=	=	SYM
ma-287	569	1	e	e	X
ma-287	569	2	−xe2(−x	−xe2(−x	NOUN
ma-287	569	3	)	)	PUNCT
ma-287	570	1	=	=	PUNCT
ma-287	571	1	e−x	e−x	NOUN
ma-287	571	2	+	+	ADJ
ma-287	571	3	∞∏	∞∏	X
ma-287	571	4	k=1	k=1	X
ma-287	571	5	2	2	NUM
ma-287	571	6	1	1	NUM
ma-287	571	7	+	+	CCONJ
ma-287	571	8	e−2−kx	e−2−kx	PROPN
ma-287	571	9	=	=	SYM
ma-287	571	10	e−x	e−x	NOUN
ma-287	571	11	[	[	PUNCT
ma-287	571	12	+	+	NOUN
ma-287	571	13	∞∏	∞∏	X
ma-287	571	14	k=1	k=1	X
ma-287	571	15	2	2	NUM
ma-287	571	16	1	1	NUM
ma-287	571	17	+	+	NUM
ma-287	571	18	e2	e2	PROPN
ma-287	571	19	−kx	−kx	NOUN
ma-287	571	20	]	]	X
ma-287	571	21	[	[	PUNCT
ma-287	571	22	+	+	NOUN
ma-287	571	23	∞∏	∞∏	X
ma-287	571	24	k=1	k=1	X
ma-287	571	25	e2	e2	PROPN
ma-287	571	26	−kx	−kx	NOUN
ma-287	571	27	]	]	PUNCT
ma-287	572	1	=	=	PUNCT
ma-287	573	1	[	[	PUNCT
ma-287	573	2	+	+	NOUN
ma-287	573	3	∞∏	∞∏	X
ma-287	573	4	k=1	k=1	X
ma-287	573	5	2	2	NUM
ma-287	573	6	1	1	NUM
ma-287	573	7	+	+	NUM
ma-287	573	8	e2	e2	PROPN
ma-287	573	9	−kx	−kx	NOUN
ma-287	573	10	]	]	PUNCT
ma-287	573	11	[	[	PUNCT
ma-287	573	12	e−x+x	e−x+x	X
ma-287	573	13	∑+∞	∑+∞	ADJ
ma-287	573	14	k=1	k=1	ADP
ma-287	573	15	2	2	NUM
ma-287	573	16	−k	−k	NOUN
ma-287	573	17	]	]	PUNCT
ma-287	574	1	=	=	PUNCT
ma-287	575	1	+	+	ADJ
ma-287	575	2	∞∏	∞∏	X
ma-287	575	3	k=1	k=1	X
ma-287	575	4	2	2	NUM
ma-287	575	5	1	1	NUM
ma-287	575	6	+	+	NUM
ma-287	575	7	e2	e2	PROPN
ma-287	575	8	−kx	−kx	NOUN
ma-287	575	9	.	.	PUNCT
ma-287	576	1	thus	thus	ADV
ma-287	576	2	,	,	PUNCT
ma-287	576	3	for	for	ADP
ma-287	576	4	x	x	PROPN
ma-287	576	5	∈	∈	PROPN
ma-287	576	6	r\{0	r\{0	PROPN
ma-287	576	7	}	}	PUNCT
ma-287	576	8	,	,	PUNCT
ma-287	576	9	the	the	DET
ma-287	576	10	formula	formula	NOUN
ma-287	576	11	in	in	ADP
ma-287	576	12	equation	equation	NOUN
ma-287	576	13	(	(	PUNCT
ma-287	576	14	4	4	X
ma-287	576	15	)	)	PUNCT
ma-287	576	16	is	be	AUX
ma-287	576	17	true	true	ADJ
ma-287	576	18	.	.	PUNCT
ma-287	577	1	to	to	ADP
ma-287	577	2	the	the	DET
ma-287	577	3	best	good	ADJ
ma-287	577	4	of	of	ADP
ma-287	577	5	our	our	PRON
ma-287	577	6	knowledge	knowledge	NOUN
ma-287	577	7	,	,	PUNCT
ma-287	577	8	this	this	DET
ma-287	577	9	specialrepresentation	specialrepresentation	NOUN
ma-287	577	10	of	of	ADP
ma-287	577	11	the	the	DET
ma-287	577	12	einstein	einstein	NOUN
ma-287	577	13	function	function	NOUN
ma-287	577	14	is	be	AUX
ma-287	577	15	a	a	DET
ma-287	577	16	new	new	ADJ
ma-287	577	17	result.the	result.the	NOUN
ma-287	577	18	next	next	ADJ
ma-287	577	19	part	part	NOUN
ma-287	577	20	is	be	AUX
ma-287	577	21	about	about	ADP
ma-287	577	22	some	some	DET
ma-287	577	23	inequalities	inequality	NOUN
ma-287	577	24	derived	derive	VERB
ma-287	577	25	from	from	ADP
ma-287	577	26	our	our	PRON
ma-287	577	27	findings	finding	NOUN
ma-287	577	28	.	.	PUNCT
ma-287	578	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	578	2	eur	eur	PROPN
ma-287	578	3	.	.	PUNCT
ma-287	579	1	j.	j.	PROPN
ma-287	579	2	math	math	PROPN
ma-287	579	3	.	.	PUNCT
ma-287	580	1	anal	anal	PROPN
ma-287	580	2	.	.	PUNCT
ma-287	581	1	10.28924	10.28924	NUM
ma-287	581	2	/	/	SYM
ma-287	581	3	ada	ada	PROPN
ma-287	581	4	/	/	SYM
ma-287	581	5	ma.5.12	ma.5.12	PROPN
ma-287	581	6	173.2	173.2	NUM
ma-287	581	7	.	.	PUNCT
ma-287	582	1	inequalities	inequality	NOUN
ma-287	582	2	.	.	PUNCT
ma-287	583	1	inequalities	inequality	NOUN
ma-287	583	2	of	of	ADP
ma-287	583	3	various	various	ADJ
ma-287	583	4	kinds	kind	NOUN
ma-287	583	5	can	can	AUX
ma-287	583	6	be	be	AUX
ma-287	583	7	derived	derive	VERB
ma-287	583	8	from	from	ADP
ma-287	583	9	our	our	PRON
ma-287	583	10	previous	previous	ADJ
ma-287	583	11	results	result	NOUN
ma-287	583	12	.	.	PUNCT
ma-287	584	1	the	the	DET
ma-287	584	2	propo	propo	NOUN
ma-287	584	3	-	-	PUNCT
ma-287	584	4	sition	sition	NOUN
ma-287	584	5	below	below	ADV
ma-287	584	6	proposes	propose	VERB
ma-287	584	7	an	an	DET
ma-287	584	8	original	original	ADJ
ma-287	584	9	one	one	NUM
ma-287	584	10	.	.	PUNCT
ma-287	585	1	proposition	proposition	NOUN
ma-287	585	2	3.2	3.2	NUM
ma-287	585	3	.	.	PUNCT
ma-287	586	1	for	for	ADP
ma-287	586	2	x	x	X
ma-287	586	3	≥	≥	NOUN
ma-287	586	4	1	1	NUM
ma-287	586	5	,	,	PUNCT
ma-287	586	6	we	we	PRON
ma-287	586	7	have	have	VERB
ma-287	586	8	√	√	NUM
ma-287	586	9	x	x	PUNCT
ma-287	587	1	−	−	PROPN
ma-287	587	2	1−	1−	NUM
ma-287	587	3	√	√	NUM
ma-287	587	4	log(x	log(x	NUM
ma-287	587	5	)	)	PUNCT
ma-287	587	6	≤	≤	NUM
ma-287	587	7	1	1	NUM
ma-287	587	8	4	4	NUM
ma-287	587	9	+	+	NOUN
ma-287	587	10	∞∑	∞∑	ADJ
ma-287	587	11	k=1	k=1	X
ma-287	587	12	2k/2(x2	2k/2(x2	PROPN
ma-287	587	13	−k	−k	PROPN
ma-287	587	14	−	−	PROPN
ma-287	588	1	1)3/2	1)3/2	NUM
ma-287	588	2	.	.	PUNCT
ma-287	589	1	this	this	DET
ma-287	589	2	last	last	ADJ
ma-287	589	3	series	series	NOUN
ma-287	589	4	expansion	expansion	NOUN
ma-287	589	5	converges	converge	VERB
ma-287	589	6	.	.	PUNCT
ma-287	590	1	proof	proof	NOUN
ma-287	590	2	.	.	PUNCT
ma-287	591	1	thanks	thank	NOUN
ma-287	591	2	to	to	ADP
ma-287	591	3	the	the	DET
ma-287	591	4	item	item	NOUN
ma-287	591	5	numbered	number	VERB
ma-287	591	6	4	4	NUM
ma-287	591	7	in	in	ADP
ma-287	591	8	proposition	proposition	NOUN
ma-287	591	9	2.2	2.2	NUM
ma-287	591	10	,	,	PUNCT
ma-287	591	11	for	for	ADP
ma-287	591	12	x	x	X
ma-287	591	13	≥	≥	NOUN
ma-287	591	14	1	1	NUM
ma-287	591	15	,	,	PUNCT
ma-287	591	16	we	we	PRON
ma-287	591	17	have	have	VERB
ma-287	591	18	√	√	NUM
ma-287	591	19	x	x	PUNCT
ma-287	592	1	−	−	PROPN
ma-287	592	2	1−	1−	NUM
ma-287	592	3	√	√	NUM
ma-287	593	1	log(x	log(x	NUM
ma-287	593	2	)	)	PUNCT
ma-287	593	3	=	=	PUNCT
ma-287	594	1	+	+	PUNCT
ma-287	594	2	∞∑	∞∑	NUM
ma-287	594	3	k=1	k=1	PUNCT
ma-287	594	4	2(k−1)/2(x2	2(k−1)/2(x2	NOUN
ma-287	594	5	−k	−k	NOUN
ma-287	594	6	−	−	PROPN
ma-287	594	7	1)3/2√	1)3/2√	NUM
ma-287	594	8	1	1	NUM
ma-287	595	1	+	+	CCONJ
ma-287	595	2	x2	x2	NOUN
ma-287	595	3	−k	−k	PROPN
ma-287	595	4	+	+	CCONJ
ma-287	595	5	√	√	NUM
ma-287	595	6	2	2	NUM
ma-287	595	7	.	.	PUNCT
ma-287	596	1	since	since	SCONJ
ma-287	596	2	x	x	X
ma-287	596	3	≥	≥	NUM
ma-287	596	4	1	1	NUM
ma-287	596	5	,	,	PUNCT
ma-287	596	6	for	for	ADP
ma-287	596	7	any	any	DET
ma-287	596	8	integer	integer	NOUN
ma-287	596	9	k	k	PROPN
ma-287	596	10	≥	≥	NUM
ma-287	596	11	1	1	NUM
ma-287	596	12	,	,	PUNCT
ma-287	596	13	we	we	PRON
ma-287	596	14	have	have	VERB
ma-287	596	15	x2−k	x2−k	PROPN
ma-287	596	16	≥	≥	NUM
ma-287	596	17	1	1	NUM
ma-287	596	18	,	,	PUNCT
ma-287	596	19	so	so	ADV
ma-287	596	20	√1	√1	PROPN
ma-287	597	1	+	+	CCONJ
ma-287	597	2	x2−k	x2−k	PROPN
ma-287	597	3	≥	≥	NOUN
ma-287	597	4	√2	√2	NOUN
ma-287	597	5	,	,	PUNCT
ma-287	597	6	which	which	PRON
ma-287	597	7	implies	imply	VERB
ma-287	597	8	that√	that√	NOUN
ma-287	597	9	1	1	NUM
ma-287	597	10	+	+	CCONJ
ma-287	597	11	x2	x2	NOUN
ma-287	597	12	−k	−k	PROPN
ma-287	597	13	+	+	CCONJ
ma-287	597	14	√	√	NUM
ma-287	597	15	2	2	NUM
ma-287	597	16	≥	≥	NOUN
ma-287	597	17	2	2	NUM
ma-287	597	18	√	√	NUM
ma-287	597	19	2	2	NUM
ma-287	597	20	.	.	PUNCT
ma-287	598	1	furthermore	furthermore	ADV
ma-287	598	2	,	,	PUNCT
ma-287	598	3	it	it	PRON
ma-287	598	4	is	be	AUX
ma-287	598	5	clear	clear	ADJ
ma-287	598	6	that	that	SCONJ
ma-287	598	7	2(k−1)/2(x2−k	2(k−1)/2(x2−k	NUM
ma-287	598	8	−	−	PROPN
ma-287	598	9	1)3/2	1)3/2	NUM
ma-287	598	10	≥	≥	NOUN
ma-287	598	11	0	0	NUM
ma-287	598	12	.	.	PUNCT
ma-287	599	1	therefore	therefore	ADV
ma-287	599	2	,	,	PUNCT
ma-287	599	3	weobtain	weobtain	NOUN
ma-287	599	4	√	√	PROPN
ma-287	599	5	x	x	PUNCT
ma-287	599	6	−	−	PROPN
ma-287	599	7	1−	1−	NUM
ma-287	599	8	√	√	NUM
ma-287	599	9	log(x	log(x	NUM
ma-287	599	10	)	)	PUNCT
ma-287	599	11	≤	≤	NUM
ma-287	599	12	1	1	NUM
ma-287	599	13	2	2	NUM
ma-287	599	14	√	√	NUM
ma-287	599	15	2	2	NUM
ma-287	599	16	+	+	NOUN
ma-287	599	17	∞∑	∞∑	ADJ
ma-287	599	18	k=1	k=1	ADP
ma-287	599	19	2(k−1)/2(x2	2(k−1)/2(x2	NUM
ma-287	599	20	−k	−k	NOUN
ma-287	599	21	−	−	PROPN
ma-287	599	22	1)3/2	1)3/2	NOUN
ma-287	600	1	=	=	NOUN
ma-287	600	2	1	1	NUM
ma-287	600	3	4	4	NUM
ma-287	600	4	+	+	NOUN
ma-287	600	5	∞∑	∞∑	ADJ
ma-287	600	6	k=1	k=1	X
ma-287	600	7	2k/2(x2	2k/2(x2	PROPN
ma-287	600	8	−k	−k	PROPN
ma-287	600	9	−	−	PROPN
ma-287	601	1	1)3/2	1)3/2	PROPN
ma-287	601	2	.	.	PUNCT
ma-287	602	1	the	the	DET
ma-287	602	2	convergence	convergence	NOUN
ma-287	602	3	of	of	ADP
ma-287	602	4	this	this	DET
ma-287	602	5	series	series	NOUN
ma-287	602	6	expansion	expansion	NOUN
ma-287	602	7	can	can	AUX
ma-287	602	8	be	be	AUX
ma-287	602	9	shown	show	VERB
ma-287	602	10	by	by	ADP
ma-287	602	11	the	the	DET
ma-287	602	12	equivalence	equivalence	NOUN
ma-287	602	13	technique	technique	NOUN
ma-287	602	14	.	.	PUNCT
ma-287	603	1	moreprecisely	moreprecisely	ADV
ma-287	603	2	,	,	PUNCT
ma-287	603	3	when	when	SCONJ
ma-287	603	4	k	k	PROPN
ma-287	603	5	→	→	SYM
ma-287	603	6	+	+	PROPN
ma-287	603	7	∞	∞	PROPN
ma-287	603	8	,	,	PUNCT
ma-287	603	9	we	we	PRON
ma-287	603	10	have	have	VERB
ma-287	603	11	2k/2(x2	2k/2(x2	NUM
ma-287	603	12	−k	−k	ADJ
ma-287	603	13	−	−	PROPN
ma-287	603	14	1)3/2	1)3/2	PROPN
ma-287	604	1	=	=	SYM
ma-287	605	1	2−k	2−k	NUM
ma-287	606	1	[	[	X
ma-287	606	2	2k(x2−k	2k(x2−k	NUM
ma-287	606	3	−	−	NOUN
ma-287	606	4	1)]3/2	1)]3/2	NUM
ma-287	606	5	∼	∼	NOUN
ma-287	606	6	2−k	2−k	NUM
ma-287	607	1	[	[	X
ma-287	607	2	log(x)]3/2	log(x)]3/2	NOUN
ma-287	607	3	,	,	PUNCT
ma-287	607	4	and	and	CCONJ
ma-287	607	5	2−k	2−k	NUM
ma-287	607	6	is	be	AUX
ma-287	607	7	the	the	DET
ma-287	607	8	term	term	NOUN
ma-287	607	9	of	of	ADP
ma-287	607	10	a	a	DET
ma-287	607	11	convergent	convergent	ADJ
ma-287	607	12	geometric	geometric	ADJ
ma-287	607	13	series	series	NOUN
ma-287	607	14	.	.	PUNCT
ma-287	608	1	the	the	DET
ma-287	608	2	desired	desire	VERB
ma-287	608	3	result	result	NOUN
ma-287	608	4	is	be	AUX
ma-287	608	5	demonstrated	demonstrate	VERB
ma-287	608	6	.	.	PUNCT
ma-287	609	1	�	�	NOUN
ma-287	609	2	this	this	DET
ma-287	609	3	inequality	inequality	NOUN
ma-287	609	4	may	may	AUX
ma-287	609	5	be	be	AUX
ma-287	609	6	more	more	ADV
ma-287	609	7	interesting	interesting	ADJ
ma-287	609	8	for	for	ADP
ma-287	609	9	the	the	DET
ma-287	609	10	lower	low	ADJ
ma-287	609	11	bound	bind	VERB
ma-287	609	12	of	of	ADP
ma-287	609	13	the	the	DET
ma-287	609	14	series	series	NOUN
ma-287	609	15	term	term	NOUN
ma-287	609	16	than	than	ADP
ma-287	609	17	for	for	ADP
ma-287	609	18	the	the	DET
ma-287	609	19	upperbound	upperbound	NOUN
ma-287	609	20	of	of	ADP
ma-287	609	21	√x	√x	ADP
ma-287	609	22	−	−	PROPN
ma-287	609	23	1−√log(x	1−√log(x	NUM
ma-287	609	24	)	)	PUNCT
ma-287	609	25	.	.	PUNCT
ma-287	610	1	in	in	ADP
ma-287	610	2	fact	fact	NOUN
ma-287	610	3	,	,	PUNCT
ma-287	610	4	it	it	PRON
ma-287	610	5	is	be	AUX
ma-287	610	6	difficult	difficult	ADJ
ma-287	610	7	to	to	PART
ma-287	610	8	capture	capture	VERB
ma-287	610	9	the	the	DET
ma-287	610	10	analytic	analytic	ADJ
ma-287	610	11	function	function	NOUN
ma-287	610	12	associated	associate	VERB
ma-287	610	13	withthe	withthe	ADJ
ma-287	610	14	series	serie	NOUN
ma-287	610	15	term.the	term.the	DET
ma-287	610	16	proposition	proposition	NOUN
ma-287	610	17	below	below	ADV
ma-287	610	18	is	be	AUX
ma-287	610	19	a	a	DET
ma-287	610	20	general	general	ADJ
ma-287	610	21	inequality	inequality	NOUN
ma-287	610	22	setting	set	VERB
ma-287	610	23	based	base	VERB
ma-287	610	24	on	on	ADP
ma-287	610	25	theorem	theorem	ADJ
ma-287	610	26	2.1	2.1	NUM
ma-287	610	27	.	.	PUNCT
ma-287	611	1	proposition	proposition	NOUN
ma-287	611	2	3.3	3.3	NUM
ma-287	611	3	.	.	PUNCT
ma-287	612	1	in	in	ADP
ma-287	612	2	the	the	DET
ma-287	612	3	framework	framework	NOUN
ma-287	612	4	of	of	ADP
ma-287	612	5	theorem	theorem	NOUN
ma-287	612	6	2.1	2.1	NUM
ma-287	612	7	,	,	PUNCT
ma-287	612	8	the	the	DET
ma-287	612	9	inequalities	inequality	NOUN
ma-287	612	10	below	below	ADV
ma-287	612	11	are	be	AUX
ma-287	612	12	true.(1	true.(1	PROPN
ma-287	612	13	)	)	PUNCT
ma-287	612	14	if	if	SCONJ
ma-287	612	15	φ	φ	PROPN
ma-287	612	16	is	be	AUX
ma-287	612	17	non	non	ADJ
ma-287	612	18	-	-	ADJ
ma-287	612	19	decreasing	decrease	VERB
ma-287	612	20	,	,	PUNCT
ma-287	612	21	then	then	ADV
ma-287	612	22	,	,	PUNCT
ma-287	612	23	for	for	ADP
ma-287	612	24	any	any	DET
ma-287	612	25	sets	set	NOUN
ma-287	612	26	of	of	ADP
ma-287	612	27	integers	integer	NOUN
ma-287	612	28	m	m	VERB
ma-287	612	29	⊆	⊆	NUM
ma-287	612	30	{	{	PUNCT
ma-287	612	31	1	1	NUM
ma-287	612	32	,	,	PUNCT
ma-287	612	33	2	2	NUM
ma-287	612	34	,	,	PUNCT
ma-287	612	35	.	.	PUNCT
ma-287	612	36	.	.	PUNCT
ma-287	613	1	.	.	PUNCT
ma-287	614	1	}	}	PUNCT
ma-287	614	2	,	,	PUNCT
ma-287	614	3	we	we	PRON
ma-287	614	4	have	have	VERB
ma-287	614	5	φ(x	φ(x	PROPN
ma-287	614	6	−	−	PROPN
ma-287	614	7	1)−	1)−	PROPN
ma-287	614	8	φ[log(x	φ[log(x	NUM
ma-287	614	9	)	)	PUNCT
ma-287	614	10	]	]	PUNCT
ma-287	614	11	≥	≥	X
ma-287	614	12	∑	∑	PUNCT
ma-287	614	13	k∈m	k∈m	ADJ
ma-287	614	14	αk(φ)(x	αk(φ)(x	PROPN
ma-287	614	15	)	)	PUNCT
ma-287	614	16	.	.	PUNCT
ma-287	615	1	(	(	PUNCT
ma-287	615	2	2	2	X
ma-287	615	3	)	)	PUNCT
ma-287	615	4	if	if	SCONJ
ma-287	615	5	φ	φ	PROPN
ma-287	615	6	is	be	AUX
ma-287	615	7	non	non	ADJ
ma-287	615	8	-	-	ADJ
ma-287	615	9	increasing	increase	VERB
ma-287	615	10	,	,	PUNCT
ma-287	615	11	then	then	ADV
ma-287	615	12	,	,	PUNCT
ma-287	615	13	for	for	ADP
ma-287	615	14	any	any	DET
ma-287	615	15	sets	set	NOUN
ma-287	615	16	of	of	ADP
ma-287	615	17	integers	integer	NOUN
ma-287	615	18	m	m	VERB
ma-287	615	19	⊆	⊆	NUM
ma-287	615	20	{	{	PUNCT
ma-287	615	21	1	1	NUM
ma-287	615	22	,	,	PUNCT
ma-287	615	23	2	2	NUM
ma-287	615	24	,	,	PUNCT
ma-287	615	25	.	.	PUNCT
ma-287	615	26	.	.	PUNCT
ma-287	616	1	.	.	PUNCT
ma-287	617	1	}	}	PUNCT
ma-287	617	2	,	,	PUNCT
ma-287	617	3	we	we	PRON
ma-287	617	4	have	have	VERB
ma-287	617	5	φ(x	φ(x	PROPN
ma-287	617	6	−	−	PROPN
ma-287	617	7	1)−	1)−	PROPN
ma-287	617	8	φ[log(x	φ[log(x	NUM
ma-287	617	9	)	)	PUNCT
ma-287	617	10	]	]	PUNCT
ma-287	617	11	≤	≤	NUM
ma-287	617	12	∑	∑	PUNCT
ma-287	617	13	k∈m	k∈m	ADJ
ma-287	617	14	αk(φ)(x	αk(φ)(x	PROPN
ma-287	617	15	)	)	PUNCT
ma-287	617	16	.	.	PUNCT
ma-287	618	1	proof	proof	NOUN
ma-287	618	2	.	.	PUNCT
ma-287	619	1	for	for	ADP
ma-287	619	2	x	x	SYM
ma-287	619	3	>	>	X
ma-287	619	4	0	0	NUM
ma-287	619	5	,	,	PUNCT
ma-287	619	6	let	let	VERB
ma-287	619	7	us	we	PRON
ma-287	619	8	consider	consider	VERB
ma-287	619	9	the	the	DET
ma-287	619	10	following	follow	VERB
ma-287	619	11	function	function	NOUN
ma-287	619	12	:	:	PUNCT
ma-287	619	13	ψ(y	ψ(y	NUM
ma-287	619	14	)	)	PUNCT
ma-287	619	15	=	=	SYM
ma-287	619	16	y(x1	y(x1	PROPN
ma-287	619	17	/	/	SYM
ma-287	619	18	y	y	PROPN
ma-287	619	19	−	−	PROPN
ma-287	619	20	1	1	NUM
ma-287	619	21	)	)	PUNCT
ma-287	619	22	,	,	PUNCT
ma-287	619	23	y	y	PROPN
ma-287	619	24	∈	∈	PROPN
ma-287	619	25	(	(	PUNCT
ma-287	619	26	0,+∞	0,+∞	NUM
ma-287	619	27	)	)	PUNCT
ma-287	619	28	.	.	PUNCT
ma-287	620	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	620	2	eur	eur	PROPN
ma-287	620	3	.	.	PUNCT
ma-287	621	1	j.	j.	PROPN
ma-287	621	2	math	math	PROPN
ma-287	621	3	.	.	PUNCT
ma-287	622	1	anal	anal	PROPN
ma-287	622	2	.	.	PUNCT
ma-287	623	1	10.28924	10.28924	NUM
ma-287	623	2	/	/	SYM
ma-287	623	3	ada	ada	PROPN
ma-287	623	4	/	/	SYM
ma-287	623	5	ma.5.12	ma.5.12	PROPN
ma-287	624	1	18then	18then	X
ma-287	624	2	we	we	PRON
ma-287	624	3	have	have	VERB
ma-287	624	4	ψ′(y	ψ′(y	VERB
ma-287	624	5	)	)	PUNCT
ma-287	625	1	=	=	SYM
ma-287	626	1	x1	x1	PROPN
ma-287	626	2	/	/	SYM
ma-287	626	3	y	y	PROPN
ma-287	626	4	−	−	PROPN
ma-287	626	5	1	1	NUM
ma-287	626	6	y	y	PROPN
ma-287	626	7	x1	x1	PROPN
ma-287	626	8	/	/	SYM
ma-287	626	9	y	y	PROPN
ma-287	626	10	log(x)−	log(x)−	PROPN
ma-287	626	11	1	1	NUM
ma-287	626	12	=	=	SYM
ma-287	626	13	x1	x1	PROPN
ma-287	626	14	/	/	SYM
ma-287	626	15	y	y	PROPN
ma-287	626	16	−	−	PROPN
ma-287	627	1	x1	x1	PROPN
ma-287	627	2	/	/	SYM
ma-287	627	3	y	y	PROPN
ma-287	627	4	log(x1	log(x1	PROPN
ma-287	627	5	/	/	SYM
ma-287	627	6	y	y	PROPN
ma-287	627	7	)	)	PUNCT
ma-287	627	8	−	−	PROPN
ma-287	628	1	1	1	X
ma-287	628	2	.	.	PUNCT
ma-287	628	3	using	use	VERB
ma-287	628	4	the	the	DET
ma-287	628	5	well	well	ADV
ma-287	628	6	-	-	PUNCT
ma-287	628	7	known	know	VERB
ma-287	628	8	logarithmic	logarithmic	ADJ
ma-287	628	9	identity	identity	NOUN
ma-287	628	10	log(t	log(t	NOUN
ma-287	628	11	)	)	PUNCT
ma-287	628	12	≥	≥	NOUN
ma-287	628	13	t−1(t	t−1(t	ADV
ma-287	628	14	−	−	NUM
ma-287	628	15	1	1	NUM
ma-287	628	16	)	)	PUNCT
ma-287	628	17	for	for	ADP
ma-287	628	18	t	t	PROPN
ma-287	628	19	>	>	X
ma-287	628	20	0	0	NUM
ma-287	628	21	,	,	PUNCT
ma-287	628	22	with	with	ADP
ma-287	628	23	t	t	NOUN
ma-287	628	24	=	=	SYM
ma-287	628	25	x1	x1	PROPN
ma-287	628	26	/	/	SYM
ma-287	628	27	y	y	PROPN
ma-287	628	28	,	,	PUNCT
ma-287	628	29	we	we	PRON
ma-287	628	30	obtain	obtain	VERB
ma-287	628	31	ψ′(y	ψ′(y	NOUN
ma-287	628	32	)	)	PUNCT
ma-287	628	33	≤	≤	NOUN
ma-287	629	1	x1	x1	PROPN
ma-287	629	2	/	/	SYM
ma-287	629	3	y	y	PROPN
ma-287	629	4	−	−	NOUN
ma-287	629	5	x1	x1	PROPN
ma-287	629	6	/	/	SYM
ma-287	629	7	yx−1	yx−1	NOUN
ma-287	629	8	/	/	SYM
ma-287	629	9	y	y	PROPN
ma-287	629	10	(	(	PUNCT
ma-287	629	11	x1	x1	PROPN
ma-287	629	12	/	/	SYM
ma-287	629	13	y	y	PROPN
ma-287	629	14	−	−	PROPN
ma-287	629	15	1)−	1)−	NUM
ma-287	629	16	1	1	NUM
ma-287	629	17	=	=	SYM
ma-287	629	18	0	0	NUM
ma-287	629	19	.	.	PUNCT
ma-287	630	1	as	as	ADP
ma-287	630	2	a	a	DET
ma-287	630	3	result	result	NOUN
ma-287	630	4	,	,	PUNCT
ma-287	630	5	ψ	ψ	X
ma-287	630	6	is	be	AUX
ma-287	630	7	a	a	DET
ma-287	630	8	non	non	ADJ
ma-287	630	9	-	-	ADJ
ma-287	630	10	increasing	increasing	ADJ
ma-287	630	11	function	function	NOUN
ma-287	630	12	.	.	PUNCT
ma-287	631	1	then	then	ADV
ma-287	631	2	,	,	PUNCT
ma-287	631	3	for	for	ADP
ma-287	631	4	any	any	DET
ma-287	631	5	integer	integer	NOUN
ma-287	631	6	k	k	PROPN
ma-287	631	7	≥	≥	NUM
ma-287	631	8	1	1	NUM
ma-287	631	9	,	,	PUNCT
ma-287	631	10	since	since	SCONJ
ma-287	631	11	2k−1	2k−1	NUM
ma-287	631	12	≤	≤	NUM
ma-287	631	13	2k	2k	NOUN
ma-287	631	14	,	,	PUNCT
ma-287	631	15	we	we	PRON
ma-287	631	16	have	have	VERB
ma-287	631	17	ψ(2k	ψ(2k	NOUN
ma-287	631	18	)	)	PUNCT
ma-287	631	19	≤	≤	NOUN
ma-287	631	20	ψ(2k−1	ψ(2k−1	NOUN
ma-287	631	21	)	)	PUNCT
ma-287	631	22	.	.	PUNCT
ma-287	632	1	let	let	VERB
ma-287	632	2	us	we	PRON
ma-287	632	3	now	now	ADV
ma-287	632	4	distinguish	distinguish	VERB
ma-287	632	5	two	two	NUM
ma-287	632	6	cases	case	NOUN
ma-287	632	7	:	:	PUNCT
ma-287	632	8	•	•	NOUN
ma-287	632	9	if	if	SCONJ
ma-287	632	10	φ	φ	PROPN
ma-287	632	11	is	be	AUX
ma-287	632	12	non	non	ADJ
ma-287	632	13	-	-	ADJ
ma-287	632	14	decreasing	decrease	VERB
ma-287	632	15	,	,	PUNCT
ma-287	632	16	then	then	ADV
ma-287	632	17	we	we	PRON
ma-287	632	18	have	have	VERB
ma-287	632	19	φ	φ	PROPN
ma-287	632	20	[	[	X
ma-287	632	21	ψ(2k	ψ(2k	X
ma-287	632	22	)	)	PUNCT
ma-287	632	23	]	]	PUNCT
ma-287	633	1	≤	≤	NUM
ma-287	633	2	φ	φ	NUM
ma-287	633	3	[	[	X
ma-287	633	4	ψ(2k−1	ψ(2k−1	NOUN
ma-287	633	5	)	)	PUNCT
ma-287	633	6	]	]	PUNCT
ma-287	633	7	,	,	PUNCT
ma-287	633	8	implying	imply	VERB
ma-287	633	9	that	that	SCONJ
ma-287	633	10	αk(φ)(x	αk(φ)(x	NOUN
ma-287	633	11	)	)	PUNCT
ma-287	634	1	=	=	SYM
ma-287	634	2	φ	φ	PROPN
ma-287	634	3	[	[	PUNCT
ma-287	634	4	ψ(2k−1	ψ(2k−1	NOUN
ma-287	634	5	)	)	PUNCT
ma-287	634	6	]	]	PUNCT
ma-287	635	1	−	−	PROPN
ma-287	635	2	φ	φ	PROPN
ma-287	635	3	[	[	PUNCT
ma-287	635	4	ψ(2k	ψ(2k	PROPN
ma-287	635	5	)	)	PUNCT
ma-287	635	6	]	]	PUNCT
ma-287	635	7	≥	≥	NOUN
ma-287	635	8	0	0	NUM
ma-287	635	9	.	.	PUNCT
ma-287	636	1	it	it	PRON
ma-287	636	2	follows	follow	VERB
ma-287	636	3	from	from	ADP
ma-287	636	4	theorem	theorem	ADJ
ma-287	636	5	2.1	2.1	NUM
ma-287	636	6	that	that	PRON
ma-287	636	7	,	,	PUNCT
ma-287	636	8	for	for	ADP
ma-287	636	9	any	any	DET
ma-287	636	10	sets	set	NOUN
ma-287	636	11	of	of	ADP
ma-287	636	12	integers	integer	NOUN
ma-287	636	13	m	m	VERB
ma-287	636	14	⊆	⊆	NUM
ma-287	636	15	{	{	PUNCT
ma-287	636	16	1	1	NUM
ma-287	636	17	,	,	PUNCT
ma-287	636	18	2	2	NUM
ma-287	636	19	,	,	PUNCT
ma-287	636	20	.	.	PUNCT
ma-287	636	21	.	.	PUNCT
ma-287	637	1	.	.	PUNCT
ma-287	638	1	}	}	PUNCT
ma-287	638	2	,	,	PUNCT
ma-287	638	3	we	we	PRON
ma-287	638	4	have	have	VERB
ma-287	638	5	φ(x	φ(x	PROPN
ma-287	638	6	−	−	PROPN
ma-287	638	7	1)−	1)−	PROPN
ma-287	638	8	φ[log(x	φ[log(x	NUM
ma-287	638	9	)	)	PUNCT
ma-287	638	10	]	]	PUNCT
ma-287	639	1	=	=	PUNCT
ma-287	640	1	+	+	PUNCT
ma-287	640	2	∞∑	∞∑	ADJ
ma-287	640	3	k=1	k=1	X
ma-287	640	4	αk(φ)(x	αk(φ)(x	NOUN
ma-287	640	5	)	)	PUNCT
ma-287	641	1	=	=	PUNCT
ma-287	641	2	∑	∑	PUNCT
ma-287	641	3	k∈m	k∈m	ADJ
ma-287	641	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	641	5	)	)	PUNCT
ma-287	642	1	+	+	CCONJ
ma-287	642	2	∑	∑	PUNCT
ma-287	642	3	k	k	PROPN
ma-287	642	4	6∈m	6∈m	NUM
ma-287	642	5	αk(φ)(x	αk(φ)(x	PROPN
ma-287	642	6	)	)	PUNCT
ma-287	642	7	≥	≥	AUX
ma-287	642	8	∑	∑	PUNCT
ma-287	642	9	k∈m	k∈m	VERB
ma-287	642	10	αk(φ)(x	αk(φ)(x	PROPN
ma-287	642	11	)	)	PUNCT
ma-287	642	12	.	.	PUNCT
ma-287	643	1	•	•	NUM
ma-287	643	2	with	with	ADP
ma-287	643	3	similar	similar	ADJ
ma-287	643	4	arguments	argument	NOUN
ma-287	643	5	,	,	PUNCT
ma-287	643	6	if	if	SCONJ
ma-287	643	7	φ	φ	PROPN
ma-287	643	8	is	be	AUX
ma-287	643	9	non	non	ADJ
ma-287	643	10	-	-	ADJ
ma-287	643	11	increasing	increase	VERB
ma-287	643	12	,	,	PUNCT
ma-287	643	13	then	then	ADV
ma-287	643	14	we	we	PRON
ma-287	643	15	have	have	VERB
ma-287	643	16	φ	φ	PROPN
ma-287	643	17	[	[	X
ma-287	643	18	ψ(2k	ψ(2k	X
ma-287	643	19	)	)	PUNCT
ma-287	643	20	]	]	PUNCT
ma-287	644	1	≥	≥	PROPN
ma-287	644	2	φ	φ	X
ma-287	644	3	[	[	PUNCT
ma-287	644	4	ψ(2k−1	ψ(2k−1	X
ma-287	644	5	)	)	PUNCT
ma-287	644	6	]	]	PUNCT
ma-287	644	7	,	,	PUNCT
ma-287	644	8	implying	imply	VERB
ma-287	644	9	that	that	SCONJ
ma-287	644	10	αk(φ)(x	αk(φ)(x	NOUN
ma-287	644	11	)	)	PUNCT
ma-287	645	1	=	=	PUNCT
ma-287	645	2	φ	φ	PROPN
ma-287	646	1	[	[	X
ma-287	646	2	ψ(2k−1)]−	ψ(2k−1)]−	PROPN
ma-287	646	3	φ	φ	PROPN
ma-287	646	4	[	[	X
ma-287	646	5	ψ(2k	ψ(2k	X
ma-287	646	6	)	)	PUNCT
ma-287	646	7	]	]	PUNCT
ma-287	647	1	≤	≤	NUM
ma-287	647	2	0	0	X
ma-287	647	3	.	.	PUNCT
ma-287	648	1	it	it	PRON
ma-287	648	2	follows	follow	VERB
ma-287	648	3	from	from	ADP
ma-287	648	4	theorem	theorem	ADJ
ma-287	648	5	2.1	2.1	NUM
ma-287	648	6	that	that	PRON
ma-287	648	7	,	,	PUNCT
ma-287	648	8	for	for	ADP
ma-287	648	9	any	any	DET
ma-287	648	10	sets	set	NOUN
ma-287	648	11	of	of	ADP
ma-287	648	12	integers	integer	NOUN
ma-287	648	13	m	m	VERB
ma-287	648	14	⊆	⊆	NUM
ma-287	648	15	{	{	PUNCT
ma-287	648	16	1	1	NUM
ma-287	648	17	,	,	PUNCT
ma-287	648	18	2	2	NUM
ma-287	648	19	,	,	PUNCT
ma-287	648	20	.	.	PUNCT
ma-287	648	21	.	.	PUNCT
ma-287	649	1	.	.	PUNCT
ma-287	650	1	}	}	PUNCT
ma-287	650	2	,	,	PUNCT
ma-287	650	3	φ(x	φ(x	PROPN
ma-287	650	4	−	−	PROPN
ma-287	650	5	1)−	1)−	PROPN
ma-287	650	6	φ[log(x	φ[log(x	NUM
ma-287	650	7	)	)	PUNCT
ma-287	650	8	]	]	PUNCT
ma-287	651	1	=	=	PUNCT
ma-287	652	1	+	+	PUNCT
ma-287	652	2	∞∑	∞∑	ADJ
ma-287	652	3	k=1	k=1	X
ma-287	652	4	αk(φ)(x	αk(φ)(x	NOUN
ma-287	652	5	)	)	PUNCT
ma-287	653	1	=	=	PUNCT
ma-287	653	2	∑	∑	PUNCT
ma-287	653	3	k∈m	k∈m	ADJ
ma-287	653	4	αk(φ)(x	αk(φ)(x	PROPN
ma-287	653	5	)	)	PUNCT
ma-287	654	1	+	+	CCONJ
ma-287	654	2	∑	∑	PUNCT
ma-287	654	3	k	k	PROPN
ma-287	654	4	6∈m	6∈m	NUM
ma-287	654	5	αk(φ)(x	αk(φ)(x	PROPN
ma-287	654	6	)	)	PUNCT
ma-287	654	7	≤	≤	NOUN
ma-287	654	8	∑	∑	PUNCT
ma-287	654	9	k∈m	k∈m	ADJ
ma-287	654	10	αk(φ)(x	αk(φ)(x	PROPN
ma-287	654	11	)	)	PUNCT
ma-287	654	12	.	.	PUNCT
ma-287	655	1	the	the	DET
ma-287	655	2	desired	desire	VERB
ma-287	655	3	inequalities	inequality	NOUN
ma-287	655	4	are	be	AUX
ma-287	655	5	demonstrated	demonstrate	VERB
ma-287	655	6	.	.	PUNCT
ma-287	656	1	�	�	PROPN
ma-287	656	2	let	let	VERB
ma-287	656	3	us	we	PRON
ma-287	656	4	exemplify	exemplify	VERB
ma-287	656	5	this	this	DET
ma-287	656	6	general	general	ADJ
ma-287	656	7	result	result	NOUN
ma-287	656	8	with	with	ADP
ma-287	656	9	an	an	DET
ma-287	656	10	immediate	immediate	ADJ
ma-287	656	11	application	application	NOUN
ma-287	656	12	.	.	PUNCT
ma-287	657	1	taking	take	VERB
ma-287	657	2	m	m	NOUN
ma-287	657	3	=	=	PUNCT
ma-287	657	4	{	{	PUNCT
ma-287	657	5	m	m	PROPN
ma-287	657	6	,	,	PUNCT
ma-287	657	7	.	.	PUNCT
ma-287	657	8	.	.	PUNCT
ma-287	658	1	.	.	PUNCT
ma-287	659	1	,	,	PUNCT
ma-287	659	2	n},where	n},where	ADV
ma-287	659	3	m	m	VERB
ma-287	659	4	and	and	CCONJ
ma-287	659	5	n	n	PRON
ma-287	659	6	are	be	AUX
ma-287	659	7	integers	integer	NOUN
ma-287	659	8	such	such	ADJ
ma-287	659	9	that	that	SCONJ
ma-287	659	10	n	n	PROPN
ma-287	659	11	≥	≥	NOUN
ma-287	659	12	m	m	VERB
ma-287	659	13	≥	≥	NOUN
ma-287	659	14	1	1	NUM
ma-287	659	15	,	,	PUNCT
ma-287	659	16	with	with	ADP
ma-287	659	17	regard	regard	NOUN
ma-287	659	18	to	to	ADP
ma-287	659	19	the	the	DET
ma-287	659	20	item	item	NOUN
ma-287	659	21	numbered	number	VERB
ma-287	659	22	3	3	NUM
ma-287	659	23	inproposition	inproposition	NOUN
ma-287	659	24	2.2	2.2	NUM
ma-287	659	25	using	use	VERB
ma-287	659	26	φ(t	φ(t	NUM
ma-287	659	27	)	)	PUNCT
ma-287	659	28	=	=	SYM
ma-287	659	29	t2	t2	NOUN
ma-287	659	30	,	,	PUNCT
ma-287	659	31	the	the	DET
ma-287	659	32	following	follow	VERB
ma-287	659	33	inequalities	inequality	NOUN
ma-287	659	34	hold	hold	VERB
ma-287	659	35	:	:	PUNCT
ma-287	659	36	•	•	NOUN
ma-287	659	37	for	for	ADP
ma-287	659	38	x	x	SYM
ma-287	659	39	>	>	X
ma-287	659	40	1	1	NUM
ma-287	659	41	,	,	PUNCT
ma-287	659	42	we	we	PRON
ma-287	659	43	have	have	VERB
ma-287	659	44	(	(	PUNCT
ma-287	659	45	x	x	SYM
ma-287	659	46	−	−	PROPN
ma-287	659	47	1)2	1)2	NUM
ma-287	659	48	−	−	PUNCT
ma-287	660	1	[	[	X
ma-287	660	2	log(x)]2	log(x)]2	X
ma-287	660	3	≥	≥	NOUN
ma-287	660	4	n∑	n∑	NOUN
ma-287	661	1	k	k	X
ma-287	662	1	=	=	NOUN
ma-287	662	2	m	m	PROPN
ma-287	662	3	22(k−1)(x2	22(k−1)(x2	NOUN
ma-287	662	4	−k	−k	NOUN
ma-287	662	5	−	−	PROPN
ma-287	662	6	1)3(3	1)3(3	PROPN
ma-287	662	7	+	+	NUM
ma-287	662	8	x2−k	x2−k	PROPN
ma-287	662	9	)	)	PUNCT
ma-287	662	10	.	.	PUNCT
ma-287	663	1	•	•	NOUN
ma-287	663	2	for	for	ADP
ma-287	663	3	x	x	PROPN
ma-287	663	4	∈	∈	PROPN
ma-287	663	5	(	(	PUNCT
ma-287	663	6	0	0	NUM
ma-287	663	7	,	,	PUNCT
ma-287	663	8	1	1	NUM
ma-287	663	9	)	)	PUNCT
ma-287	663	10	,	,	PUNCT
ma-287	663	11	we	we	PRON
ma-287	663	12	have	have	VERB
ma-287	663	13	(	(	PUNCT
ma-287	663	14	x	x	SYM
ma-287	663	15	−	−	PROPN
ma-287	663	16	1)2	1)2	NUM
ma-287	663	17	−	−	PUNCT
ma-287	664	1	[	[	X
ma-287	664	2	log(x)]2	log(x)]2	X
ma-287	664	3	≤	≤	NUM
ma-287	664	4	n∑	n∑	NOUN
ma-287	665	1	k	k	X
ma-287	665	2	=	=	PROPN
ma-287	665	3	m	m	PROPN
ma-287	665	4	22(k−1)(x2	22(k−1)(x2	NOUN
ma-287	665	5	−k	−k	NOUN
ma-287	665	6	−	−	PROPN
ma-287	665	7	1)3(3	1)3(3	PROPN
ma-287	665	8	+	+	NUM
ma-287	665	9	x2−k	x2−k	PROPN
ma-287	665	10	)	)	PUNCT
ma-287	665	11	.	.	PUNCT
ma-287	666	1	we	we	PRON
ma-287	666	2	can	can	AUX
ma-287	666	3	also	also	ADV
ma-287	666	4	remark	remark	VERB
ma-287	666	5	that	that	SCONJ
ma-287	666	6	,	,	PUNCT
ma-287	666	7	for	for	ADP
ma-287	666	8	any	any	DET
ma-287	666	9	integer	integer	NOUN
ma-287	666	10	k	k	PROPN
ma-287	666	11	≥	≥	NUM
ma-287	666	12	1	1	NUM
ma-287	666	13	,	,	PUNCT
ma-287	666	14	if	if	SCONJ
ma-287	666	15	x	x	PROPN
ma-287	666	16	>	>	X
ma-287	666	17	1	1	NUM
ma-287	666	18	,	,	PUNCT
ma-287	666	19	then	then	ADV
ma-287	666	20	we	we	PRON
ma-287	666	21	have	have	VERB
ma-287	666	22	(	(	PUNCT
ma-287	666	23	x2−k	x2−k	PROPN
ma-287	666	24	−	−	PROPN
ma-287	667	1	1)3	1)3	PROPN
ma-287	667	2	>	>	X
ma-287	667	3	0	0	NUM
ma-287	667	4	,	,	PUNCT
ma-287	667	5	and	and	CCONJ
ma-287	667	6	if	if	SCONJ
ma-287	667	7	x	x	X
ma-287	667	8	∈	∈	PROPN
ma-287	667	9	(	(	PUNCT
ma-287	667	10	0	0	NUM
ma-287	667	11	,	,	PUNCT
ma-287	667	12	1	1	NUM
ma-287	667	13	)	)	PUNCT
ma-287	667	14	,	,	PUNCT
ma-287	667	15	then	then	ADV
ma-287	667	16	we	we	PRON
ma-287	667	17	have	have	VERB
ma-287	667	18	(	(	PUNCT
ma-287	668	1	x2−k	x2−k	PROPN
ma-287	668	2	−	−	PROPN
ma-287	669	1	1)3	1)3	PROPN
ma-287	669	2	<	<	X
ma-287	669	3	0	0	NUM
ma-287	669	4	,	,	PUNCT
ma-287	669	5	and	and	CCONJ
ma-287	669	6	the	the	DET
ma-287	669	7	above	above	ADJ
ma-287	669	8	inequalities	inequalitie	VERB
ma-287	669	9	follow.another	follow.another	PRON
ma-287	669	10	simple	simple	ADJ
ma-287	669	11	application	application	NOUN
ma-287	669	12	is	be	AUX
ma-287	669	13	the	the	DET
ma-287	669	14	inequality	inequality	NOUN
ma-287	669	15	formulated	formulate	VERB
ma-287	669	16	in	in	ADP
ma-287	669	17	the	the	DET
ma-287	669	18	lemma	lemma	PROPN
ma-287	669	19	below	below	ADV
ma-287	669	20	.	.	PUNCT
ma-287	670	1	lemma	lemma	PROPN
ma-287	670	2	3.4	3.4	NUM
ma-287	670	3	.	.	PUNCT
ma-287	671	1	for	for	ADP
ma-287	671	2	x	x	SYM
ma-287	671	3	>	>	X
ma-287	671	4	0	0	NUM
ma-287	671	5	,	,	PUNCT
ma-287	671	6	we	we	PRON
ma-287	671	7	have	have	VERB
ma-287	671	8	log(x	log(x	NUM
ma-287	671	9	)	)	PUNCT
ma-287	671	10	≤	≤	NOUN
ma-287	671	11	2	2	NUM
ma-287	671	12	(	(	PUNCT
ma-287	671	13	√	√	NUM
ma-287	671	14	x	x	SYM
ma-287	671	15	−	−	PROPN
ma-287	671	16	1	1	NUM
ma-287	671	17	)	)	PUNCT
ma-287	671	18	.	.	PUNCT
ma-287	672	1	proof	proof	NOUN
ma-287	672	2	.	.	PUNCT
ma-287	673	1	we	we	PRON
ma-287	673	2	propose	propose	VERB
ma-287	673	3	three	three	NUM
ma-287	673	4	different	different	ADJ
ma-287	673	5	proofs	proof	NOUN
ma-287	673	6	.	.	PUNCT
ma-287	674	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	674	2	eur	eur	PROPN
ma-287	674	3	.	.	PUNCT
ma-287	675	1	j.	j.	PROPN
ma-287	675	2	math	math	PROPN
ma-287	675	3	.	.	PUNCT
ma-287	676	1	anal	anal	PROPN
ma-287	676	2	.	.	PUNCT
ma-287	677	1	10.28924	10.28924	NUM
ma-287	677	2	/	/	SYM
ma-287	677	3	ada	ada	PROPN
ma-287	677	4	/	/	SYM
ma-287	677	5	ma.5.12	ma.5.12	PROPN
ma-287	677	6	19	19	NUM
ma-287	677	7	proof	proof	NOUN
ma-287	677	8	1	1	NUM
ma-287	677	9	:	:	PUNCT
ma-287	677	10	judicious	judicious	ADJ
ma-287	677	11	approach	approach	NOUN
ma-287	677	12	.	.	PUNCT
ma-287	678	1	:	:	PUNCT
ma-287	678	2	the	the	DET
ma-287	678	3	following	follow	VERB
ma-287	678	4	inequality	inequality	NOUN
ma-287	678	5	is	be	AUX
ma-287	678	6	well	well	ADV
ma-287	678	7	known	know	VERB
ma-287	678	8	:	:	PUNCT
ma-287	678	9	log(y	log(y	PROPN
ma-287	678	10	)	)	PUNCT
ma-287	678	11	≤	≤	NOUN
ma-287	679	1	y	y	NOUN
ma-287	679	2	−	−	PROPN
ma-287	679	3	1	1	NUM
ma-287	679	4	for	for	ADP
ma-287	679	5	y	y	PROPN
ma-287	679	6	>	>	X
ma-287	679	7	0	0	X
ma-287	679	8	.	.	PUNCT
ma-287	679	9	applying	apply	VERB
ma-287	679	10	it	it	PRON
ma-287	679	11	to	to	ADP
ma-287	679	12	the	the	DET
ma-287	679	13	judicious	judicious	ADJ
ma-287	679	14	choice	choice	NOUN
ma-287	679	15	y	y	PROPN
ma-287	679	16	=	=	PUNCT
ma-287	679	17	√x	√x	NOUN
ma-287	679	18	,	,	PUNCT
ma-287	679	19	we	we	PRON
ma-287	679	20	get	get	VERB
ma-287	679	21	log(x	log(x	NUM
ma-287	679	22	)	)	PUNCT
ma-287	679	23	=	=	SYM
ma-287	679	24	2	2	NUM
ma-287	679	25	log(√x	log(√x	VERB
ma-287	679	26	)	)	PUNCT
ma-287	679	27	≤	≤	NOUN
ma-287	679	28	2(√x−1	2(√x−1	NUM
ma-287	679	29	)	)	PUNCT
ma-287	679	30	.	.	PUNCT
ma-287	680	1	proof	proof	NOUN
ma-287	680	2	2	2	NUM
ma-287	680	3	:	:	PUNCT
ma-287	680	4	use	use	NOUN
ma-287	680	5	of	of	ADP
ma-287	680	6	our	our	PRON
ma-287	680	7	series	series	NOUN
ma-287	680	8	expansion	expansion	NOUN
ma-287	680	9	.	.	PUNCT
ma-287	681	1	:	:	PUNCT
ma-287	681	2	we	we	PRON
ma-287	681	3	can	can	AUX
ma-287	681	4	apply	apply	VERB
ma-287	681	5	the	the	DET
ma-287	681	6	first	first	ADJ
ma-287	681	7	items	item	NOUN
ma-287	681	8	in	in	ADP
ma-287	681	9	propositions	proposition	NOUN
ma-287	681	10	2.2	2.2	NUM
ma-287	681	11	and3.3	and3.3	PROPN
ma-287	681	12	with	with	ADP
ma-287	681	13	m	m	NOUN
ma-287	681	14	=	=	PUNCT
ma-287	681	15	{	{	PUNCT
ma-287	681	16	1	1	NUM
ma-287	681	17	}	}	PUNCT
ma-287	681	18	.	.	PUNCT
ma-287	682	1	indeed	indeed	ADV
ma-287	682	2	,	,	PUNCT
ma-287	682	3	since	since	SCONJ
ma-287	682	4	φ(t	φ(t	PROPN
ma-287	682	5	)	)	PUNCT
ma-287	682	6	is	be	AUX
ma-287	682	7	non	non	ADJ
ma-287	682	8	-	-	ADJ
ma-287	682	9	decreasing	decrease	VERB
ma-287	682	10	(	(	PUNCT
ma-287	682	11	or	or	CCONJ
ma-287	682	12	the	the	DET
ma-287	682	13	coefficients	coefficient	NOUN
ma-287	682	14	of	of	ADP
ma-287	682	15	the	the	DET
ma-287	682	16	relatedseries	relatedserie	NOUN
ma-287	682	17	expansion	expansion	NOUN
ma-287	682	18	are	be	AUX
ma-287	682	19	clearly	clearly	ADV
ma-287	682	20	non	non	ADJ
ma-287	682	21	-	-	ADJ
ma-287	682	22	negative	negative	ADJ
ma-287	682	23	)	)	PUNCT
ma-287	682	24	,	,	PUNCT
ma-287	682	25	we	we	PRON
ma-287	682	26	have	have	VERB
ma-287	682	27	x	x	X
ma-287	682	28	−	−	PROPN
ma-287	682	29	1−	1−	NUM
ma-287	682	30	log(x	log(x	NUM
ma-287	682	31	)	)	PUNCT
ma-287	682	32	≥	≥	NOUN
ma-287	683	1	∑	∑	PUNCT
ma-287	683	2	k∈m	k∈m	VERB
ma-287	683	3	2k−1(x2	2k−1(x2	NOUN
ma-287	683	4	−k	−k	NOUN
ma-287	683	5	−	−	X
ma-287	683	6	1)2	1)2	NUM
ma-287	683	7	=	=	SYM
ma-287	683	8	(	(	PUNCT
ma-287	683	9	√	√	INTJ
ma-287	683	10	x	x	PUNCT
ma-287	683	11	−	−	PROPN
ma-287	683	12	1)2	1)2	NUM
ma-287	683	13	,	,	PUNCT
ma-287	683	14	implying	imply	VERB
ma-287	683	15	that	that	SCONJ
ma-287	683	16	log(x	log(x	NOUN
ma-287	683	17	)	)	PUNCT
ma-287	683	18	≤	≤	NOUN
ma-287	683	19	x	x	PUNCT
ma-287	684	1	−	−	PROPN
ma-287	684	2	1−	1−	NUM
ma-287	685	1	(	(	PUNCT
ma-287	685	2	√	√	NUM
ma-287	685	3	x	x	PUNCT
ma-287	685	4	−	−	PROPN
ma-287	685	5	1)2	1)2	NUM
ma-287	685	6	=	=	SYM
ma-287	685	7	2	2	NUM
ma-287	685	8	(	(	PUNCT
ma-287	685	9	√	√	NUM
ma-287	685	10	x	x	SYM
ma-287	685	11	−	−	PROPN
ma-287	685	12	1	1	NUM
ma-287	685	13	)	)	PUNCT
ma-287	685	14	.	.	PUNCT
ma-287	686	1	proof	proof	NOUN
ma-287	686	2	3	3	NUM
ma-287	686	3	:	:	PUNCT
ma-287	686	4	use	use	NOUN
ma-287	686	5	of	of	ADP
ma-287	686	6	differentiation	differentiation	NOUN
ma-287	686	7	.	.	PUNCT
ma-287	687	1	:	:	PUNCT
ma-287	687	2	in	in	ADP
ma-287	687	3	the	the	DET
ma-287	687	4	proof	proof	NOUN
ma-287	687	5	of	of	ADP
ma-287	687	6	proposition	proposition	NOUN
ma-287	687	7	3.3	3.3	NUM
ma-287	687	8	,	,	PUNCT
ma-287	687	9	we	we	PRON
ma-287	687	10	showed	show	VERB
ma-287	687	11	that	that	SCONJ
ma-287	687	12	,	,	PUNCT
ma-287	687	13	for	for	ADP
ma-287	687	14	x	x	SYM
ma-287	687	15	>	>	X
ma-287	687	16	0,the	0,the	PRON
ma-287	687	17	function	function	NOUN
ma-287	687	18	ψ(y	ψ(y	NOUN
ma-287	687	19	)	)	PUNCT
ma-287	687	20	=	=	SYM
ma-287	687	21	y(x1	y(x1	PROPN
ma-287	687	22	/	/	SYM
ma-287	687	23	y	y	PROPN
ma-287	687	24	−	−	PROPN
ma-287	687	25	1	1	NUM
ma-287	687	26	)	)	PUNCT
ma-287	687	27	is	be	AUX
ma-287	687	28	non	non	ADJ
ma-287	687	29	-	-	ADJ
ma-287	687	30	increasing	increasing	ADJ
ma-287	687	31	.	.	PUNCT
ma-287	688	1	this	this	PRON
ma-287	688	2	implies	imply	VERB
ma-287	688	3	that	that	SCONJ
ma-287	688	4	,	,	PUNCT
ma-287	688	5	for	for	ADP
ma-287	688	6	any	any	DET
ma-287	688	7	θ	θ	PROPN
ma-287	688	8	>	>	X
ma-287	688	9	0	0	PROPN
ma-287	688	10	,	,	PUNCT
ma-287	688	11	wehave	wehave	NOUN
ma-287	688	12	limy→0	limy→0	PROPN
ma-287	688	13	+	+	CCONJ
ma-287	688	14	ψ(y	ψ(y	NOUN
ma-287	688	15	)	)	PUNCT
ma-287	688	16	≤	≤	NOUN
ma-287	688	17	ψ(θ	ψ(θ	NOUN
ma-287	688	18	)	)	PUNCT
ma-287	688	19	,	,	PUNCT
ma-287	688	20	i.e.	i.e.	X
ma-287	688	21	,	,	PUNCT
ma-287	688	22	log(x	log(x	NUM
ma-287	688	23	)	)	PUNCT
ma-287	688	24	≤	≤	NOUN
ma-287	688	25	θ(x1	θ(x1	ADJ
ma-287	688	26	/	/	SYM
ma-287	688	27	θ	θ	PROPN
ma-287	688	28	−	−	NOUN
ma-287	688	29	1	1	NUM
ma-287	688	30	)	)	PUNCT
ma-287	688	31	.	.	PUNCT
ma-287	689	1	the	the	DET
ma-287	689	2	desired	desire	VERB
ma-287	689	3	result	result	NOUN
ma-287	689	4	is	be	AUX
ma-287	689	5	just	just	ADV
ma-287	689	6	a	a	DET
ma-287	689	7	special	special	ADJ
ma-287	689	8	case	case	NOUN
ma-287	689	9	;	;	PUNCT
ma-287	689	10	it	it	PRON
ma-287	689	11	is	be	AUX
ma-287	689	12	enough	enough	ADJ
ma-287	689	13	to	to	PART
ma-287	689	14	take	take	VERB
ma-287	689	15	θ	θ	NOUN
ma-287	689	16	=	=	SYM
ma-287	689	17	2.this	2.this	NUM
ma-287	689	18	completes	complete	VERB
ma-287	689	19	the	the	DET
ma-287	689	20	proof	proof	NOUN
ma-287	689	21	.	.	PUNCT
ma-287	690	1	�	�	PROPN
ma-287	691	1	this	this	DET
ma-287	691	2	lemma	lemma	PROPN
ma-287	691	3	is	be	AUX
ma-287	691	4	not	not	PART
ma-287	691	5	new	new	ADJ
ma-287	691	6	;	;	PUNCT
ma-287	691	7	it	it	PRON
ma-287	691	8	has	have	AUX
ma-287	691	9	been	be	AUX
ma-287	691	10	demonstrated	demonstrate	VERB
ma-287	691	11	with	with	ADP
ma-287	691	12	other	other	ADJ
ma-287	691	13	differentiation	differentiation	NOUN
ma-287	691	14	techniques	technique	NOUN
ma-287	691	15	in	in	ADP
ma-287	691	16	[	[	X
ma-287	691	17	17],and	17],and	NUM
ma-287	691	18	its	its	PRON
ma-287	691	19	sharpness	sharpness	NOUN
ma-287	691	20	has	have	AUX
ma-287	691	21	also	also	ADV
ma-287	691	22	been	be	AUX
ma-287	691	23	illustrated.another	illustrated.another	DET
ma-287	691	24	logarithmic	logarithmic	ADJ
ma-287	691	25	inequality	inequality	NOUN
ma-287	691	26	is	be	AUX
ma-287	691	27	highlighted	highlight	VERB
ma-287	691	28	in	in	ADP
ma-287	691	29	the	the	DET
ma-287	691	30	lemma	lemma	PROPN
ma-287	691	31	below	below	ADV
ma-287	691	32	.	.	PUNCT
ma-287	692	1	lemma	lemma	PROPN
ma-287	692	2	3.5	3.5	NUM
ma-287	692	3	.	.	PUNCT
ma-287	693	1	for	for	ADP
ma-287	693	2	x	x	SYM
ma-287	693	3	>	>	X
ma-287	693	4	0	0	NUM
ma-287	693	5	and	and	CCONJ
ma-287	693	6	any	any	DET
ma-287	693	7	integers	integer	NOUN
ma-287	693	8	m	m	VERB
ma-287	693	9	and	and	CCONJ
ma-287	693	10	n	n	CCONJ
ma-287	693	11	such	such	ADJ
ma-287	693	12	that	that	SCONJ
ma-287	693	13	n	n	CCONJ
ma-287	693	14	≥	≥	NOUN
ma-287	693	15	m	m	VERB
ma-287	693	16	≥	≥	NUM
ma-287	693	17	1	1	NUM
ma-287	693	18	,	,	PUNCT
ma-287	693	19	we	we	PRON
ma-287	693	20	have	have	VERB
ma-287	693	21	log(x	log(x	NUM
ma-287	693	22	)	)	PUNCT
ma-287	693	23	≤	≤	NOUN
ma-287	694	1	(	(	PUNCT
ma-287	694	2	x	x	SYM
ma-287	694	3	−	−	NOUN
ma-287	694	4	1	1	NUM
ma-287	694	5	)	)	PUNCT
ma-287	694	6	n∏	n∏	NOUN
ma-287	694	7	k	k	NOUN
ma-287	694	8	=	=	NOUN
ma-287	694	9	m	m	VERB
ma-287	694	10	2	2	NUM
ma-287	694	11	1	1	NUM
ma-287	694	12	+	+	CCONJ
ma-287	694	13	x2	x2	PROPN
ma-287	694	14	−k	−k	ADJ
ma-287	694	15	.	.	PUNCT
ma-287	695	1	proof	proof	NOUN
ma-287	695	2	.	.	PUNCT
ma-287	696	1	the	the	DET
ma-287	696	2	proof	proof	NOUN
ma-287	696	3	is	be	AUX
ma-287	696	4	a	a	DET
ma-287	696	5	consequence	consequence	NOUN
ma-287	696	6	of	of	ADP
ma-287	696	7	the	the	DET
ma-287	696	8	infinite	infinite	ADJ
ma-287	696	9	product	product	NOUN
ma-287	696	10	expansion	expansion	NOUN
ma-287	696	11	in	in	ADP
ma-287	696	12	equation	equation	NOUN
ma-287	696	13	(	(	PUNCT
ma-287	696	14	3	3	NUM
ma-287	696	15	)	)	PUNCT
ma-287	696	16	.	.	PUNCT
ma-287	697	1	let	let	VERB
ma-287	697	2	us	we	PRON
ma-287	697	3	distin	distin	VERB
ma-287	697	4	-	-	PUNCT
ma-287	697	5	guish	guish	NOUN
ma-287	697	6	the	the	DET
ma-287	697	7	cases	case	NOUN
ma-287	697	8	x	x	PUNCT
ma-287	697	9	≥	≥	NUM
ma-287	697	10	1	1	NUM
ma-287	697	11	and	and	CCONJ
ma-287	697	12	x	x	SYM
ma-287	697	13	∈	∈	PROPN
ma-287	697	14	(	(	PUNCT
ma-287	697	15	0	0	NUM
ma-287	697	16	,	,	PUNCT
ma-287	697	17	1	1	NUM
ma-287	697	18	)	)	PUNCT
ma-287	697	19	.	.	PUNCT
ma-287	698	1	•	•	NOUN
ma-287	698	2	for	for	ADP
ma-287	698	3	x	x	X
ma-287	698	4	≥	≥	NOUN
ma-287	698	5	1	1	NUM
ma-287	698	6	and	and	CCONJ
ma-287	698	7	any	any	DET
ma-287	698	8	integer	integer	NOUN
ma-287	698	9	k	k	PROPN
ma-287	698	10	≥	≥	NUM
ma-287	698	11	1	1	NUM
ma-287	698	12	,	,	PUNCT
ma-287	698	13	we	we	PRON
ma-287	698	14	have	have	VERB
ma-287	698	15	x2−k	x2−k	PROPN
ma-287	698	16	≥	≥	NUM
ma-287	698	17	1	1	NUM
ma-287	698	18	,	,	PUNCT
ma-287	698	19	implying	imply	VERB
ma-287	698	20	that	that	SCONJ
ma-287	698	21	2/(1	2/(1	PROPN
ma-287	698	22	+	+	CCONJ
ma-287	698	23	x2−k	x2−k	PROPN
ma-287	698	24	)	)	PUNCT
ma-287	699	1	≤	≤	NUM
ma-287	700	1	1	1	NUM
ma-287	700	2	.	.	PUNCT
ma-287	701	1	sowe	sowe	NOUN
ma-287	701	2	have	have	VERB
ma-287	701	3	log(x	log(x	PROPN
ma-287	701	4	)	)	PUNCT
ma-287	701	5	=	=	SYM
ma-287	702	1	(	(	PUNCT
ma-287	702	2	x	x	SYM
ma-287	702	3	−	−	PROPN
ma-287	702	4	1	1	NUM
ma-287	702	5	)	)	PUNCT
ma-287	702	6	[	[	PUNCT
ma-287	702	7	m−1∏	m−1∏	ADJ
ma-287	702	8	k=1	k=1	NOUN
ma-287	702	9	2	2	NUM
ma-287	702	10	1	1	NUM
ma-287	702	11	+	+	CCONJ
ma-287	702	12	x2	x2	NOUN
ma-287	702	13	−k	−k	NOUN
ma-287	702	14	]	]	X
ma-287	702	15	[	[	PUNCT
ma-287	702	16	n∏	n∏	NOUN
ma-287	702	17	k	k	NOUN
ma-287	702	18	=	=	NOUN
ma-287	702	19	m	m	VERB
ma-287	702	20	2	2	NUM
ma-287	702	21	1	1	NUM
ma-287	702	22	+	+	CCONJ
ma-287	702	23	x2	x2	NOUN
ma-287	702	24	−k	−k	PROPN
ma-287	702	25	]	]	X
ma-287	702	26	[	[	PUNCT
ma-287	702	27	+	+	ADJ
ma-287	702	28	∞∏	∞∏	X
ma-287	702	29	k	k	NOUN
ma-287	702	30	=	=	NOUN
ma-287	702	31	n+1	n+1	PROPN
ma-287	702	32	2	2	NUM
ma-287	702	33	1	1	NUM
ma-287	702	34	+	+	CCONJ
ma-287	702	35	x2	x2	PROPN
ma-287	702	36	−k	−k	NOUN
ma-287	702	37	]	]	PUNCT
ma-287	702	38	≤	≤	NUM
ma-287	702	39	(	(	PUNCT
ma-287	702	40	x	x	SYM
ma-287	702	41	−	−	NOUN
ma-287	702	42	1	1	NUM
ma-287	702	43	)	)	PUNCT
ma-287	702	44	n∏	n∏	NOUN
ma-287	702	45	k	k	NOUN
ma-287	703	1	=	=	NOUN
ma-287	703	2	m	m	VERB
ma-287	703	3	2	2	NUM
ma-287	703	4	1	1	NUM
ma-287	703	5	+	+	CCONJ
ma-287	703	6	x2	x2	NOUN
ma-287	703	7	−k	−k	PROPN
ma-287	703	8	,	,	PUNCT
ma-287	703	9	with	with	ADP
ma-287	703	10	the	the	DET
ma-287	703	11	convention	convention	NOUN
ma-287	703	12	∏0k=1[2/(1	∏0k=1[2/(1	PROPN
ma-287	703	13	+	+	CCONJ
ma-287	703	14	x2−k	x2−k	PROPN
ma-287	703	15	)	)	PUNCT
ma-287	703	16	]	]	PUNCT
ma-287	704	1	=	=	PUNCT
ma-287	704	2	1	1	X
ma-287	704	3	.	.	NOUN
ma-287	704	4	•	•	NUM
ma-287	704	5	for	for	ADP
ma-287	704	6	x	x	PROPN
ma-287	704	7	∈	∈	PROPN
ma-287	704	8	(	(	PUNCT
ma-287	704	9	0	0	NUM
ma-287	704	10	,	,	PUNCT
ma-287	704	11	1	1	NUM
ma-287	704	12	)	)	PUNCT
ma-287	704	13	and	and	CCONJ
ma-287	704	14	any	any	DET
ma-287	704	15	integer	integer	NOUN
ma-287	704	16	k	k	PROPN
ma-287	704	17	≥	≥	NUM
ma-287	704	18	1	1	NUM
ma-287	704	19	,	,	PUNCT
ma-287	704	20	we	we	PRON
ma-287	704	21	have	have	VERB
ma-287	704	22	x2−k	x2−k	PROPN
ma-287	704	23	≤	≤	NUM
ma-287	704	24	1	1	NUM
ma-287	704	25	,	,	PUNCT
ma-287	704	26	implying	imply	VERB
ma-287	704	27	that	that	SCONJ
ma-287	704	28	2/(1	2/(1	PROPN
ma-287	704	29	+	+	CCONJ
ma-287	704	30	x2−k	x2−k	PROPN
ma-287	704	31	)	)	PUNCT
ma-287	704	32	≥	≥	PROPN
ma-287	705	1	1and	1and	NUM
ma-287	705	2	x	x	SYM
ma-287	705	3	−	−	PROPN
ma-287	705	4	1	1	NUM
ma-287	705	5	≤	≤	NOUN
ma-287	705	6	0	0	NUM
ma-287	705	7	.	.	PUNCT
ma-287	706	1	the	the	DET
ma-287	706	2	exact	exact	ADJ
ma-287	706	3	same	same	ADJ
ma-287	706	4	inequality	inequality	NOUN
ma-287	706	5	as	as	ADP
ma-287	706	6	above	above	ADV
ma-287	706	7	is	be	AUX
ma-287	706	8	obtained.the	obtained.the	DET
ma-287	706	9	proof	proof	NOUN
ma-287	706	10	is	be	AUX
ma-287	706	11	therefore	therefore	ADV
ma-287	706	12	finished	finish	VERB
ma-287	706	13	.	.	PUNCT
ma-287	707	1	�	�	PROPN
ma-287	707	2	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	707	3	eur	eur	PROPN
ma-287	707	4	.	.	PUNCT
ma-287	708	1	j.	j.	PROPN
ma-287	708	2	math	math	PROPN
ma-287	708	3	.	.	PUNCT
ma-287	709	1	anal	anal	PROPN
ma-287	709	2	.	.	PUNCT
ma-287	710	1	10.28924	10.28924	NUM
ma-287	710	2	/	/	SYM
ma-287	710	3	ada	ada	PROPN
ma-287	710	4	/	/	SYM
ma-287	710	5	ma.5.12	ma.5.12	PROPN
ma-287	710	6	20this	20this	DET
ma-287	710	7	lemma	lemma	PROPN
ma-287	710	8	generalizes	generalize	VERB
ma-287	710	9	the	the	DET
ma-287	710	10	famous	famous	ADJ
ma-287	710	11	inequality	inequality	NOUN
ma-287	710	12	log(x	log(x	PROPN
ma-287	710	13	)	)	PUNCT
ma-287	710	14	≤	≤	NUM
ma-287	710	15	x	x	PUNCT
ma-287	711	1	−	−	PROPN
ma-287	711	2	1	1	NUM
ma-287	711	3	for	for	ADP
ma-287	711	4	x	x	PUNCT
ma-287	711	5	>	>	X
ma-287	711	6	0	0	X
ma-287	711	7	.	.	PUNCT
ma-287	712	1	it	it	PRON
ma-287	712	2	also	also	ADV
ma-287	712	3	shows	show	VERB
ma-287	712	4	how	how	SCONJ
ma-287	712	5	theseidel	theseidel	NOUN
ma-287	712	6	formula	formula	NOUN
ma-287	712	7	,	,	PUNCT
ma-287	712	8	i.e.	i.e.	X
ma-287	712	9	,	,	PUNCT
ma-287	712	10	equation	equation	NOUN
ma-287	712	11	(	(	PUNCT
ma-287	712	12	3	3	NUM
ma-287	712	13	)	)	PUNCT
ma-287	712	14	,	,	PUNCT
ma-287	712	15	can	can	AUX
ma-287	712	16	be	be	AUX
ma-287	712	17	applied	apply	VERB
ma-287	712	18	to	to	PART
ma-287	712	19	refine	refine	VERB
ma-287	712	20	it.a	it.a	ADJ
ma-287	712	21	similar	similar	ADJ
ma-287	712	22	inequality	inequality	NOUN
ma-287	712	23	involving	involve	VERB
ma-287	712	24	the	the	DET
ma-287	712	25	einstein	einstein	ADJ
ma-287	712	26	function	function	NOUN
ma-287	712	27	is	be	AUX
ma-287	712	28	examined	examine	VERB
ma-287	712	29	below	below	ADV
ma-287	712	30	.	.	PUNCT
ma-287	713	1	lemma	lemma	PROPN
ma-287	713	2	3.6	3.6	NUM
ma-287	713	3	.	.	PUNCT
ma-287	714	1	for	for	ADP
ma-287	714	2	x	x	SYM
ma-287	714	3	>	>	X
ma-287	714	4	0	0	NUM
ma-287	714	5	and	and	CCONJ
ma-287	714	6	any	any	DET
ma-287	714	7	integers	integer	NOUN
ma-287	714	8	m	m	VERB
ma-287	714	9	and	and	CCONJ
ma-287	714	10	n	n	CCONJ
ma-287	714	11	such	such	ADJ
ma-287	714	12	that	that	SCONJ
ma-287	714	13	n	n	CCONJ
ma-287	714	14	≥	≥	NOUN
ma-287	714	15	m	m	VERB
ma-287	714	16	≥	≥	NUM
ma-287	714	17	1	1	NUM
ma-287	714	18	,	,	PUNCT
ma-287	714	19	we	we	PRON
ma-287	714	20	have	have	VERB
ma-287	714	21	e2(x	e2(x	NOUN
ma-287	714	22	)	)	PUNCT
ma-287	715	1	≤	≤	PROPN
ma-287	715	2	n∏	n∏	PROPN
ma-287	715	3	k	k	PROPN
ma-287	716	1	=	=	NOUN
ma-287	716	2	m	m	VERB
ma-287	716	3	2	2	NUM
ma-287	716	4	1	1	NUM
ma-287	716	5	+	+	NUM
ma-287	716	6	e2	e2	PROPN
ma-287	716	7	−kx	−kx	NOUN
ma-287	716	8	.	.	PUNCT
ma-287	717	1	for	for	ADP
ma-287	717	2	x	x	SYM
ma-287	717	3	<	<	X
ma-287	717	4	0	0	NUM
ma-287	717	5	,	,	PUNCT
ma-287	717	6	the	the	DET
ma-287	717	7	reversed	reverse	VERB
ma-287	717	8	inequality	inequality	NOUN
ma-287	717	9	holds	hold	VERB
ma-287	717	10	.	.	PUNCT
ma-287	718	1	proof	proof	NOUN
ma-287	718	2	.	.	PUNCT
ma-287	719	1	the	the	DET
ma-287	719	2	proof	proof	NOUN
ma-287	719	3	follows	follow	VERB
ma-287	719	4	from	from	ADP
ma-287	719	5	the	the	DET
ma-287	719	6	infinite	infinite	ADJ
ma-287	719	7	product	product	NOUN
ma-287	719	8	expansion	expansion	NOUN
ma-287	719	9	in	in	ADP
ma-287	719	10	equation	equation	NOUN
ma-287	719	11	(	(	PUNCT
ma-287	719	12	4	4	NUM
ma-287	719	13	)	)	PUNCT
ma-287	719	14	.	.	PUNCT
ma-287	720	1	indeed	indeed	ADV
ma-287	720	2	,	,	PUNCT
ma-287	720	3	for	for	ADP
ma-287	720	4	x	x	SYM
ma-287	720	5	>	>	X
ma-287	720	6	0	0	PUNCT
ma-287	720	7	andany	andany	ADJ
ma-287	720	8	integer	integer	NOUN
ma-287	720	9	k	k	PROPN
ma-287	720	10	≥	≥	NUM
ma-287	720	11	1	1	NUM
ma-287	720	12	,	,	PUNCT
ma-287	720	13	we	we	PRON
ma-287	720	14	have	have	VERB
ma-287	720	15	e2−kx	e2−kx	NOUN
ma-287	720	16	>	>	SYM
ma-287	720	17	1	1	NUM
ma-287	720	18	,	,	PUNCT
ma-287	720	19	implying	imply	VERB
ma-287	720	20	that	that	SCONJ
ma-287	720	21	2/(1	2/(1	NUM
ma-287	720	22	+	+	CCONJ
ma-287	720	23	e2−kx	e2−kx	NOUN
ma-287	720	24	)	)	PUNCT
ma-287	720	25	<	<	X
ma-287	721	1	1	1	X
ma-287	721	2	.	.	PUNCT
ma-287	721	3	as	as	ADP
ma-287	721	4	a	a	DET
ma-287	721	5	result	result	NOUN
ma-287	721	6	,	,	PUNCT
ma-287	721	7	we	we	PRON
ma-287	721	8	have	have	VERB
ma-287	721	9	e2(x	e2(x	NOUN
ma-287	721	10	)	)	PUNCT
ma-287	721	11	=	=	NOUN
ma-287	722	1	[	[	PUNCT
ma-287	722	2	m−1∏	m−1∏	ADJ
ma-287	722	3	k=1	k=1	NOUN
ma-287	722	4	2	2	NUM
ma-287	722	5	1	1	NUM
ma-287	722	6	+	+	NUM
ma-287	722	7	e2	e2	PROPN
ma-287	722	8	−kx	−kx	NOUN
ma-287	722	9	]	]	X
ma-287	722	10	[	[	PUNCT
ma-287	722	11	n∏	n∏	PROPN
ma-287	722	12	k	k	NOUN
ma-287	722	13	=	=	NOUN
ma-287	722	14	m	m	VERB
ma-287	722	15	2	2	NUM
ma-287	722	16	1	1	NUM
ma-287	722	17	+	+	NUM
ma-287	722	18	e2	e2	PROPN
ma-287	722	19	−kx	−kx	NOUN
ma-287	722	20	]	]	X
ma-287	722	21	[	[	PUNCT
ma-287	722	22	+	+	ADJ
ma-287	722	23	∞∏	∞∏	X
ma-287	722	24	k	k	NOUN
ma-287	722	25	=	=	NOUN
ma-287	722	26	n+1	n+1	PROPN
ma-287	722	27	2	2	NUM
ma-287	722	28	1	1	NUM
ma-287	722	29	+	+	NUM
ma-287	722	30	e2	e2	PROPN
ma-287	722	31	−kx	−kx	NOUN
ma-287	722	32	]	]	PUNCT
ma-287	722	33	≤	≤	PROPN
ma-287	722	34	n∏	n∏	PROPN
ma-287	722	35	k	k	PROPN
ma-287	723	1	=	=	NOUN
ma-287	723	2	m	m	VERB
ma-287	723	3	2	2	NUM
ma-287	723	4	1	1	NUM
ma-287	723	5	+	+	NUM
ma-287	723	6	e2	e2	PROPN
ma-287	723	7	−kx	−kx	NOUN
ma-287	723	8	,	,	PUNCT
ma-287	723	9	with	with	ADP
ma-287	723	10	the	the	DET
ma-287	723	11	convention	convention	NOUN
ma-287	723	12	∏0k=1[2/(1	∏0k=1[2/(1	PROPN
ma-287	723	13	+	+	CCONJ
ma-287	723	14	e2−kx	e2−kx	NUM
ma-287	723	15	)	)	PUNCT
ma-287	723	16	]	]	PUNCT
ma-287	724	1	=	=	PUNCT
ma-287	724	2	1	1	X
ma-287	724	3	.	.	X
ma-287	725	1	for	for	ADP
ma-287	725	2	x	x	SYM
ma-287	725	3	<	<	X
ma-287	725	4	0	0	NUM
ma-287	725	5	,	,	PUNCT
ma-287	725	6	for	for	AUX
ma-287	725	7	x	x	X
ma-287	725	8	>	>	X
ma-287	725	9	0	0	NUM
ma-287	725	10	and	and	CCONJ
ma-287	725	11	any	any	DET
ma-287	725	12	integer	integer	NOUN
ma-287	725	13	k	k	PROPN
ma-287	725	14	≥	≥	NUM
ma-287	725	15	1	1	NUM
ma-287	725	16	,	,	PUNCT
ma-287	725	17	wehave	wehave	NOUN
ma-287	725	18	e2−kx	e2−kx	ADP
ma-287	725	19	<	<	X
ma-287	725	20	1	1	NUM
ma-287	725	21	,	,	PUNCT
ma-287	725	22	implying	imply	VERB
ma-287	725	23	that	that	SCONJ
ma-287	725	24	2/(1	2/(1	NUM
ma-287	725	25	+	+	CCONJ
ma-287	725	26	e2−kx	e2−kx	NOUN
ma-287	725	27	)	)	PUNCT
ma-287	725	28	>	>	X
ma-287	725	29	1	1	X
ma-287	725	30	.	.	PUNCT
ma-287	725	31	using	use	VERB
ma-287	725	32	this	this	PRON
ma-287	725	33	,	,	PUNCT
ma-287	725	34	the	the	DET
ma-287	725	35	reversed	reversed	ADJ
ma-287	725	36	inequality	inequality	NOUN
ma-287	725	37	above	above	ADP
ma-287	725	38	isimmediately	isimmediately	ADV
ma-287	725	39	established	establish	VERB
ma-287	725	40	.	.	PUNCT
ma-287	726	1	the	the	DET
ma-287	726	2	desired	desire	VERB
ma-287	726	3	results	result	NOUN
ma-287	726	4	are	be	AUX
ma-287	726	5	obtained	obtain	VERB
ma-287	726	6	.	.	PUNCT
ma-287	727	1	�	�	PROPN
ma-287	727	2	the	the	DET
ma-287	727	3	inequalities	inequality	NOUN
ma-287	727	4	above	above	ADV
ma-287	727	5	are	be	AUX
ma-287	727	6	just	just	ADV
ma-287	727	7	a	a	DET
ma-287	727	8	sample	sample	NOUN
ma-287	727	9	of	of	ADP
ma-287	727	10	what	what	PRON
ma-287	727	11	can	can	AUX
ma-287	727	12	be	be	AUX
ma-287	727	13	derived	derive	VERB
ma-287	727	14	from	from	ADP
ma-287	727	15	our	our	PRON
ma-287	727	16	results	result	NOUN
ma-287	727	17	;	;	PUNCT
ma-287	727	18	other	other	ADJ
ma-287	727	19	explo	explo	NOUN
ma-287	727	20	-	-	PUNCT
ma-287	727	21	rations	ration	NOUN
ma-287	727	22	are	be	AUX
ma-287	727	23	left	leave	VERB
ma-287	727	24	for	for	ADP
ma-287	727	25	future	future	ADJ
ma-287	727	26	studies	study	NOUN
ma-287	727	27	.	.	PUNCT
ma-287	728	1	4	4	X
ma-287	728	2	.	.	X
ma-287	728	3	conclusion	conclusion	NOUN
ma-287	728	4	in	in	ADP
ma-287	728	5	conclusion	conclusion	NOUN
ma-287	728	6	,	,	PUNCT
ma-287	728	7	the	the	DET
ma-287	728	8	logarithmic	logarithmic	ADJ
ma-287	728	9	function	function	NOUN
ma-287	728	10	log(x	log(x	PROPN
ma-287	728	11	)	)	PUNCT
ma-287	728	12	is	be	AUX
ma-287	728	13	central	central	ADJ
ma-287	728	14	to	to	ADP
ma-287	728	15	several	several	ADJ
ma-287	728	16	mathematical	mathematical	ADJ
ma-287	728	17	disciplines	discipline	NOUN
ma-287	728	18	.	.	PUNCT
ma-287	729	1	inparticular	inparticular	PROPN
ma-287	729	2	,	,	PUNCT
ma-287	729	3	its	its	PRON
ma-287	729	4	series	series	NOUN
ma-287	729	5	expansions	expansion	NOUN
ma-287	729	6	play	play	VERB
ma-287	729	7	a	a	DET
ma-287	729	8	crucial	crucial	ADJ
ma-287	729	9	role	role	NOUN
ma-287	729	10	in	in	ADP
ma-287	729	11	mathematical	mathematical	ADJ
ma-287	729	12	analysis	analysis	NOUN
ma-287	729	13	.	.	PUNCT
ma-287	730	1	while	while	SCONJ
ma-287	730	2	the	the	DET
ma-287	730	3	classicalseries	classicalserie	NOUN
ma-287	730	4	expansion	expansion	NOUN
ma-287	730	5	initially	initially	ADV
ma-287	730	6	presented	present	VERB
ma-287	730	7	limitations	limitation	NOUN
ma-287	730	8	for	for	ADP
ma-287	730	9	the	the	DET
ma-287	730	10	natural	natural	ADJ
ma-287	730	11	domain	domain	NOUN
ma-287	730	12	,	,	PUNCT
ma-287	730	13	i.e.	i.e.	X
ma-287	730	14	,	,	PUNCT
ma-287	730	15	x	x	SYM
ma-287	730	16	∈	∈	PROPN
ma-287	730	17	(	(	PUNCT
ma-287	730	18	0,+∞	0,+∞	NUM
ma-287	730	19	)	)	PUNCT
ma-287	730	20	,	,	PUNCT
ma-287	730	21	a	a	DET
ma-287	730	22	re	re	VERB
ma-287	730	23	-	-	VERB
ma-287	730	24	fined	fined	ADJ
ma-287	730	25	telescoping	telescoping	NOUN
ma-287	730	26	technique	technique	NOUN
ma-287	730	27	elaborated	elaborate	VERB
ma-287	730	28	in	in	ADP
ma-287	730	29	[	[	X
ma-287	730	30	4	4	NUM
ma-287	730	31	]	]	PUNCT
ma-287	730	32	allows	allow	VERB
ma-287	730	33	to	to	PART
ma-287	730	34	relax	relax	VERB
ma-287	730	35	the	the	DET
ma-287	730	36	constraint	constraint	NOUN
ma-287	730	37	.	.	PUNCT
ma-287	731	1	this	this	PRON
ma-287	731	2	allows	allow	VERB
ma-287	731	3	a	a	DET
ma-287	731	4	broaderunderstanding	broaderunderstanding	NOUN
ma-287	731	5	of	of	ADP
ma-287	731	6	the	the	DET
ma-287	731	7	logarithmic	logarithmic	ADJ
ma-287	731	8	function	function	NOUN
ma-287	731	9	.	.	PUNCT
ma-287	732	1	in	in	ADP
ma-287	732	2	our	our	PRON
ma-287	732	3	first	first	ADJ
ma-287	732	4	investigations	investigation	NOUN
ma-287	732	5	,	,	PUNCT
ma-287	732	6	based	base	VERB
ma-287	732	7	on	on	ADP
ma-287	732	8	this	this	DET
ma-287	732	9	result	result	NOUN
ma-287	732	10	and	and	CCONJ
ma-287	732	11	awell	awell	NOUN
ma-287	732	12	-	-	PUNCT
ma-287	732	13	known	know	VERB
ma-287	732	14	series	series	NOUN
ma-287	732	15	expansion	expansion	NOUN
ma-287	732	16	established	establish	VERB
ma-287	732	17	by	by	ADP
ma-287	732	18	srinivasa	srinivasa	PROPN
ma-287	732	19	ramanujan	ramanujan	PROPN
ma-287	732	20	(	(	PUNCT
ma-287	732	21	see	see	VERB
ma-287	732	22	[	[	X
ma-287	732	23	2	2	NUM
ma-287	732	24	]	]	PUNCT
ma-287	732	25	and	and	CCONJ
ma-287	732	26	[	[	X
ma-287	732	27	5	5	NUM
ma-287	732	28	]	]	NUM
ma-287	732	29	)	)	PUNCT
ma-287	732	30	,	,	PUNCT
ma-287	732	31	we	we	PRON
ma-287	732	32	discovereda	discovereda	VERB
ma-287	732	33	unified	unify	VERB
ma-287	732	34	functional	functional	ADJ
ma-287	732	35	pattern	pattern	NOUN
ma-287	732	36	.	.	PUNCT
ma-287	733	1	this	this	DET
ma-287	733	2	connection	connection	NOUN
ma-287	733	3	can	can	AUX
ma-287	733	4	be	be	AUX
ma-287	733	5	expressed	express	VERB
ma-287	733	6	in	in	ADP
ma-287	733	7	the	the	DET
ma-287	733	8	form	form	NOUN
ma-287	733	9	"	"	PUNCT
ma-287	733	10	φ(x	φ(x	PROPN
ma-287	733	11	−	−	PROPN
ma-287	733	12	1	1	NUM
ma-287	733	13	)	)	PUNCT
ma-287	733	14	−	−	PROPN
ma-287	733	15	φ[log(x)]".in	φ[log(x)]".in	ADJ
ma-287	733	16	light	light	NOUN
ma-287	733	17	of	of	ADP
ma-287	733	18	this	this	PRON
ma-287	733	19	,	,	PUNCT
ma-287	733	20	using	use	VERB
ma-287	733	21	telescoping	telescope	VERB
ma-287	733	22	techniques	technique	NOUN
ma-287	733	23	and	and	CCONJ
ma-287	733	24	thorough	thorough	ADJ
ma-287	733	25	factorization	factorization	NOUN
ma-287	733	26	developments	development	NOUN
ma-287	733	27	,	,	PUNCT
ma-287	733	28	we	we	PRON
ma-287	733	29	generatenew	generatenew	NOUN
ma-287	733	30	series	series	NOUN
ma-287	733	31	extensions	extension	NOUN
ma-287	733	32	for	for	ADP
ma-287	733	33	such	such	ADJ
ma-287	733	34	functions	function	NOUN
ma-287	733	35	.	.	PUNCT
ma-287	734	1	several	several	ADJ
ma-287	734	2	examples	example	NOUN
ma-287	734	3	are	be	AUX
ma-287	734	4	given	give	VERB
ma-287	734	5	and	and	CCONJ
ma-287	734	6	discussed	discuss	VERB
ma-287	734	7	.	.	PUNCT
ma-287	735	1	as	as	SCONJ
ma-287	735	2	illustrated	illustrate	VERB
ma-287	735	3	,	,	PUNCT
ma-287	735	4	these	these	DET
ma-287	735	5	results	result	NOUN
ma-287	735	6	lead	lead	VERB
ma-287	735	7	to	to	ADP
ma-287	735	8	new	new	ADJ
ma-287	735	9	product	product	NOUN
ma-287	735	10	expansions	expansion	NOUN
ma-287	735	11	,	,	PUNCT
ma-287	735	12	including	include	VERB
ma-287	735	13	one	one	NUM
ma-287	735	14	for	for	ADP
ma-287	735	15	the	the	DET
ma-287	735	16	einstein	einstein	PROPN
ma-287	735	17	function	function	NOUN
ma-287	735	18	,	,	PUNCT
ma-287	735	19	and	and	CCONJ
ma-287	735	20	to	to	ADP
ma-287	735	21	in	in	ADP
ma-287	735	22	-	-	PUNCT
ma-287	735	23	equalities	equality	NOUN
ma-287	735	24	involving	involve	VERB
ma-287	735	25	the	the	DET
ma-287	735	26	logarithmic	logarithmic	ADJ
ma-287	735	27	function	function	NOUN
ma-287	735	28	.	.	PUNCT
ma-287	736	1	this	this	DET
ma-287	736	2	article	article	NOUN
ma-287	736	3	thus	thus	ADV
ma-287	736	4	contributes	contribute	VERB
ma-287	736	5	to	to	ADP
ma-287	736	6	a	a	DET
ma-287	736	7	better	well	ADV
ma-287	736	8	understandingof	understandingof	ADJ
ma-287	736	9	the	the	DET
ma-287	736	10	logarithmic	logarithmic	ADJ
ma-287	736	11	function	function	NOUN
ma-287	736	12	and	and	CCONJ
ma-287	736	13	series	series	NOUN
ma-287	736	14	expansions	expansion	NOUN
ma-287	736	15	in	in	ADP
ma-287	736	16	general	general	ADJ
ma-287	736	17	,	,	PUNCT
ma-287	736	18	and	and	CCONJ
ma-287	736	19	lays	lay	VERB
ma-287	736	20	some	some	DET
ma-287	736	21	foundations	foundation	NOUN
ma-287	736	22	for	for	ADP
ma-287	736	23	futurework	futurework	NOUN
ma-287	736	24	.	.	PUNCT
ma-287	737	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	737	2	eur	eur	PROPN
ma-287	737	3	.	.	PUNCT
ma-287	738	1	j.	j.	PROPN
ma-287	738	2	math	math	PROPN
ma-287	738	3	.	.	PUNCT
ma-287	739	1	anal	anal	PROPN
ma-287	739	2	.	.	PUNCT
ma-287	740	1	10.28924	10.28924	NUM
ma-287	740	2	/	/	SYM
ma-287	740	3	ada	ada	PROPN
ma-287	740	4	/	/	SYM
ma-287	740	5	ma.5.12	ma.5.12	PROPN
ma-287	740	6	21references	21references	NUM
ma-287	741	1	[	[	X
ma-287	741	2	1	1	NUM
ma-287	741	3	]	]	PUNCT
ma-287	741	4	m.	m.	NOUN
ma-287	741	5	abramowitz	abramowitz	PROPN
ma-287	741	6	,	,	PUNCT
ma-287	741	7	i.a	i.a	PROPN
ma-287	741	8	.	.	PROPN
ma-287	741	9	stegun	stegun	PROPN
ma-287	741	10	,	,	PUNCT
ma-287	741	11	"	"	PUNCT
ma-287	741	12	debye	debye	ADJ
ma-287	741	13	functions	function	NOUN
ma-287	741	14	.	.	PUNCT
ma-287	741	15	"	"	PUNCT
ma-287	742	1	27.1	27.1	NUM
ma-287	742	2	in	in	ADP
ma-287	742	3	handbook	handbook	NOUN
ma-287	742	4	of	of	ADP
ma-287	742	5	mathematical	mathematical	ADJ
ma-287	742	6	functions	function	NOUN
ma-287	742	7	with	with	ADP
ma-287	742	8	formulas	formula	NOUN
ma-287	742	9	,	,	PUNCT
ma-287	742	10	graphs	graph	NOUN
ma-287	742	11	,	,	PUNCT
ma-287	742	12	and	and	CCONJ
ma-287	742	13	mathematical	mathematical	ADJ
ma-287	742	14	tables	table	NOUN
ma-287	742	15	,	,	PUNCT
ma-287	742	16	9th	9th	ADJ
ma-287	742	17	printing	printing	NOUN
ma-287	742	18	.	.	PUNCT
ma-287	743	1	new	new	PROPN
ma-287	743	2	york	york	PROPN
ma-287	743	3	:	:	PUNCT
ma-287	743	4	dover	dover	PROPN
ma-287	743	5	,	,	PUNCT
ma-287	743	6	999	999	NUM
ma-287	743	7	-	-	SYM
ma-287	743	8	1000	1000	NUM
ma-287	743	9	,	,	PUNCT
ma-287	743	10	1972.[2	1972.[2	NUM
ma-287	743	11	]	]	X
ma-287	743	12	b.c	b.c	PROPN
ma-287	743	13	.	.	PROPN
ma-287	743	14	berndt	berndt	PROPN
ma-287	743	15	,	,	PUNCT
ma-287	743	16	ramanujan	ramanujan	PROPN
ma-287	743	17	’s	’s	PART
ma-287	743	18	notebooks	notebook	NOUN
ma-287	743	19	,	,	PUNCT
ma-287	743	20	part	part	NOUN
ma-287	743	21	iv	iv	NUM
ma-287	743	22	,	,	PUNCT
ma-287	743	23	springer	springer	NOUN
ma-287	743	24	,	,	PUNCT
ma-287	743	25	new	new	PROPN
ma-287	743	26	york	york	PROPN
ma-287	743	27	,	,	PUNCT
ma-287	743	28	1994	1994	NUM
ma-287	743	29	.	.	PUNCT
ma-287	744	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
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ma-287	744	3	-	-	SYM
ma-287	744	4	1	1	NUM
ma-287	744	5	-	-	PUNCT
ma-287	744	6	4612	4612	NUM
ma-287	744	7	-	-	PUNCT
ma-287	744	8	0879	0879	NUM
ma-287	744	9	-	-	SYM
ma-287	744	10	2.[3	2.[3	NUM
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ma-287	744	13	bougoffa	bougoffa	PROPN
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ma-287	744	15	p.	p.	PROPN
ma-287	744	16	krasopoulos	krasopoulos	PROPN
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ma-287	744	19	optimal	optimal	ADJ
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ma-287	744	22	logarithmic	logarithmic	ADJ
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ma-287	744	24	exponential	exponential	ADJ
ma-287	744	25	functions	function	NOUN
ma-287	744	26	,	,	PUNCT
ma-287	744	27	journal	journal	NOUN
ma-287	744	28	of	of	ADP
ma-287	744	29	inequalitiesand	inequalitiesand	PROPN
ma-287	744	30	special	special	ADJ
ma-287	744	31	functions	function	NOUN
ma-287	744	32	,	,	PUNCT
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ma-287	744	34	,	,	PUNCT
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ma-287	744	36	-	-	SYM
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ma-287	744	46	infinite	infinite	ADJ
ma-287	744	47	series	series	NOUN
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ma-287	744	51	concavity	concavity	NOUN
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ma-287	744	54	natural	natural	ADJ
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ma-287	746	2	(	(	PUNCT
ma-287	746	3	2017	2017	NUM
ma-287	746	4	)	)	PUNCT
ma-287	746	5	,	,	PUNCT
ma-287	746	6	353	353	NUM
ma-287	746	7	-	-	SYM
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ma-287	746	9	.	.	PUNCT
ma-287	747	1	https://doi.org/10.4169/math.mag.90.5.353.[5	https://doi.org/10.4169/math.mag.90.5.353.[5	PROPN
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ma-287	747	3	d.m	d.m	PROPN
ma-287	747	4	.	.	PROPN
ma-287	747	5	bradley	bradley	PROPN
ma-287	747	6	,	,	PUNCT
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ma-287	747	8	an	an	DET
ma-287	747	9	infinite	infinite	ADJ
ma-287	747	10	series	series	NOUN
ma-287	747	11	of	of	ADP
ma-287	747	12	ramanujan	ramanujan	PROPN
ma-287	747	13	related	relate	VERB
ma-287	747	14	to	to	ADP
ma-287	747	15	the	the	DET
ma-287	747	16	natural	natural	ADJ
ma-287	747	17	logarithm	logarithm	NOUN
ma-287	747	18	,	,	PUNCT
ma-287	747	19	ramanujan	ramanujan	NOUN
ma-287	747	20	j.	j.	PROPN
ma-287	747	21	47	47	NUM
ma-287	747	22	(	(	PUNCT
ma-287	747	23	2018),253	2018),253	NUM
ma-287	747	24	-	-	SYM
ma-287	747	25	265	265	NUM
ma-287	747	26	.	.	PUNCT
ma-287	748	1	https://doi.org/10.1007/s11139-017-9961-y.[6	https://doi.org/10.1007/s11139-017-9961-y.[6	PRON
ma-287	748	2	]	]	PUNCT
ma-287	748	3	f.	f.	PROPN
ma-287	748	4	burk	burk	PROPN
ma-287	748	5	,	,	PUNCT
ma-287	748	6	the	the	DET
ma-287	748	7	geometric	geometric	ADJ
ma-287	748	8	,	,	PUNCT
ma-287	748	9	logarithmic	logarithmic	ADJ
ma-287	748	10	,	,	PUNCT
ma-287	748	11	and	and	CCONJ
ma-287	748	12	arithmetic	arithmetic	ADJ
ma-287	748	13	mean	mean	NOUN
ma-287	748	14	inequality	inequality	NOUN
ma-287	748	15	,	,	PUNCT
ma-287	748	16	am	be	AUX
ma-287	748	17	.	.	PUNCT
ma-287	748	18	math	math	NOUN
ma-287	748	19	.	.	PUNCT
ma-287	749	1	mon	mon	PROPN
ma-287	749	2	.	.	PUNCT
ma-287	750	1	94	94	NUM
ma-287	750	2	(	(	PUNCT
ma-287	750	3	1987	1987	NUM
ma-287	750	4	)	)	PUNCT
ma-287	750	5	,	,	PUNCT
ma-287	750	6	527	527	NUM
ma-287	750	7	-	-	SYM
ma-287	750	8	528	528	NUM
ma-287	750	9	.	.	PUNCT
ma-287	751	1	https	https	NOUN
ma-287	751	2	:	:	PUNCT
ma-287	751	3	//doi.org/10.1080/00029890.1987.12000678.[7	//doi.org/10.1080/00029890.1987.12000678.[7	PUNCT
ma-287	751	4	]	]	X
ma-287	751	5	c.	c.	PROPN
ma-287	751	6	chesneau	chesneau	PROPN
ma-287	751	7	,	,	PUNCT
ma-287	751	8	y.j	y.j	PROPN
ma-287	751	9	.	.	PROPN
ma-287	751	10	bagul	bagul	NOUN
ma-287	751	11	,	,	PUNCT
ma-287	751	12	new	new	ADJ
ma-287	751	13	sharp	sharp	ADJ
ma-287	751	14	bounds	bound	NOUN
ma-287	751	15	for	for	ADP
ma-287	751	16	the	the	DET
ma-287	751	17	logarithmic	logarithmic	ADJ
ma-287	751	18	function	function	NOUN
ma-287	751	19	,	,	PUNCT
ma-287	751	20	elec	elec	PROPN
ma-287	751	21	.	.	PUNCT
ma-287	752	1	j.	j.	PROPN
ma-287	752	2	math	math	PROPN
ma-287	752	3	.	.	PUNCT
ma-287	753	1	anal	anal	PROPN
ma-287	753	2	.	.	PUNCT
ma-287	754	1	appl	appl	PROPN
ma-287	754	2	.	.	PROPN
ma-287	755	1	8	8	NUM
ma-287	755	2	(	(	PUNCT
ma-287	755	3	2020	2020	NUM
ma-287	755	4	)	)	PUNCT
ma-287	755	5	,	,	PUNCT
ma-287	755	6	140	140	NUM
ma-287	755	7	-	-	SYM
ma-287	755	8	145	145	NUM
ma-287	755	9	.	.	PUNCT
ma-287	756	1	https://doi.org/10.21608/ejmaa.2020.312813.[8	https://doi.org/10.21608/ejmaa.2020.312813.[8	X
ma-287	756	2	]	]	X
ma-287	756	3	i.s	i.s	PROPN
ma-287	756	4	.	.	PROPN
ma-287	756	5	gradshteyn	gradshteyn	PROPN
ma-287	756	6	,	,	PUNCT
ma-287	756	7	i.m	i.m	PROPN
ma-287	756	8	.	.	PROPN
ma-287	756	9	ryzhik	ryzhik	PROPN
ma-287	756	10	,	,	PUNCT
ma-287	756	11	table	table	NOUN
ma-287	756	12	of	of	ADP
ma-287	756	13	integrals	integral	NOUN
ma-287	756	14	,	,	PUNCT
ma-287	756	15	series	series	NOUN
ma-287	756	16	,	,	PUNCT
ma-287	756	17	and	and	CCONJ
ma-287	756	18	products	product	NOUN
ma-287	756	19	;	;	PUNCT
ma-287	756	20	translated	translate	VERB
ma-287	756	21	from	from	ADP
ma-287	756	22	the	the	DET
ma-287	756	23	russian	russian	NOUN
ma-287	756	24	,	,	PUNCT
ma-287	756	25	eighth	eighth	ADJ
ma-287	756	26	edition	edition	NOUN
ma-287	756	27	,	,	PUNCT
ma-287	756	28	revised	revise	VERB
ma-287	756	29	from	from	ADP
ma-287	756	30	the	the	DET
ma-287	756	31	seventh	seventh	ADJ
ma-287	756	32	edition	edition	NOUN
ma-287	756	33	;	;	PUNCT
ma-287	756	34	zwillinger	zwillinger	NOUN
ma-287	756	35	,	,	PUNCT
ma-287	756	36	d.	d.	PROPN
ma-287	756	37	,	,	PUNCT
ma-287	756	38	moll	moll	PROPN
ma-287	756	39	,	,	PUNCT
ma-287	756	40	v.	v.	PROPN
ma-287	756	41	,	,	PUNCT
ma-287	756	42	translators	translator	NOUN
ma-287	756	43	;	;	PUNCT
ma-287	756	44	elsevier	elsevier	NOUN
ma-287	756	45	/	/	SYM
ma-287	756	46	academic	academic	ADJ
ma-287	756	47	press	press	NOUN
ma-287	756	48	:	:	PUNCT
ma-287	757	1	amsterdam	amsterdam	PROPN
ma-287	757	2	,	,	PUNCT
ma-287	757	3	thenetherlands	thenetherland	NOUN
ma-287	757	4	,	,	PUNCT
ma-287	757	5	2015.[9	2015.[9	NUM
ma-287	757	6	]	]	X
ma-287	757	7	g.	g.	PROPN
ma-287	757	8	jameson	jameson	PROPN
ma-287	757	9	,	,	PUNCT
ma-287	757	10	p.r	p.r	PROPN
ma-287	757	11	.	.	PROPN
ma-287	757	12	mercer	mercer	PROPN
ma-287	757	13	,	,	PUNCT
ma-287	757	14	the	the	DET
ma-287	757	15	logarithmic	logarithmic	ADJ
ma-287	757	16	mean	mean	NOUN
ma-287	757	17	revisited	revisit	VERB
ma-287	757	18	,	,	PUNCT
ma-287	757	19	am	be	AUX
ma-287	757	20	.	.	PUNCT
ma-287	757	21	math	math	NOUN
ma-287	757	22	.	.	PUNCT
ma-287	758	1	mon	mon	PROPN
ma-287	758	2	.	.	PUNCT
ma-287	759	1	126	126	NUM
ma-287	759	2	(	(	PUNCT
ma-287	759	3	2019	2019	NUM
ma-287	759	4	)	)	PUNCT
ma-287	759	5	,	,	PUNCT
ma-287	759	6	641	641	NUM
ma-287	759	7	-	-	SYM
ma-287	759	8	645	645	NUM
ma-287	759	9	.	.	PUNCT
ma-287	760	1	https://doi.org/	https://doi.org/	PROPN
ma-287	760	2	10.1080/00029890.2019.1605799.[10	10.1080/00029890.2019.1605799.[10	PROPN
ma-287	760	3	]	]	PUNCT
ma-287	760	4	m.	m.	PROPN
ma-287	760	5	kostić	kostić	PROPN
ma-287	760	6	,	,	PUNCT
ma-287	760	7	new	new	ADJ
ma-287	760	8	inequalities	inequality	NOUN
ma-287	760	9	for	for	ADP
ma-287	760	10	the	the	DET
ma-287	760	11	function	function	NOUN
ma-287	760	12	y	y	PROPN
ma-287	760	13	=	=	SYM
ma-287	760	14	t	t	PROPN
ma-287	760	15	ln	ln	PROPN
ma-287	760	16	t	t	PROPN
ma-287	760	17	,	,	PUNCT
ma-287	760	18	electronic	electronic	ADJ
ma-287	760	19	journal	journal	NOUN
ma-287	760	20	of	of	ADP
ma-287	760	21	mathematical	mathematical	ADJ
ma-287	760	22	analysis	analysis	NOUN
ma-287	760	23	and	and	CCONJ
ma-287	760	24	applications,8	applications,8	X
ma-287	760	25	(	(	PUNCT
ma-287	760	26	2020	2020	NUM
ma-287	760	27	)	)	PUNCT
ma-287	760	28	,	,	PUNCT
ma-287	760	29	291	291	NUM
ma-287	760	30	-	-	SYM
ma-287	760	31	296	296	NUM
ma-287	760	32	.	.	PUNCT
ma-287	761	1	https://doi.org/10.21608/ejmaa.2020.312859.[11	https://doi.org/10.21608/ejmaa.2020.312859.[11	PROPN
ma-287	761	2	]	]	PUNCT
ma-287	761	3	e.r	e.r	PROPN
ma-287	761	4	.	.	PROPN
ma-287	761	5	love	love	PROPN
ma-287	761	6	,	,	PUNCT
ma-287	761	7	some	some	DET
ma-287	761	8	logarithm	logarithm	NOUN
ma-287	761	9	inequalities	inequality	NOUN
ma-287	761	10	,	,	PUNCT
ma-287	761	11	math	math	NOUN
ma-287	761	12	.	.	PUNCT
ma-287	762	1	gaz	gaz	PROPN
ma-287	762	2	.	.	PROPN
ma-287	763	1	64	64	NUM
ma-287	763	2	(	(	PUNCT
ma-287	763	3	1980	1980	NUM
ma-287	763	4	)	)	PUNCT
ma-287	763	5	,	,	PUNCT
ma-287	763	6	55	55	NUM
ma-287	763	7	-	-	SYM
ma-287	763	8	57	57	NUM
ma-287	763	9	.	.	PUNCT
ma-287	764	1	https://doi.org/10.2307/3615890.[12	https://doi.org/10.2307/3615890.[12	NOUN
ma-287	764	2	]	]	X
ma-287	764	3	s.	s.	PROPN
ma-287	764	4	ramanujan	ramanujan	PROPN
ma-287	764	5	,	,	PUNCT
ma-287	764	6	notebooks	notebook	NOUN
ma-287	764	7	(	(	PUNCT
ma-287	764	8	2	2	NUM
ma-287	764	9	volumes	volume	NOUN
ma-287	764	10	)	)	PUNCT
ma-287	764	11	,	,	PUNCT
ma-287	764	12	tata	tata	PROPN
ma-287	764	13	institute	institute	PROPN
ma-287	764	14	of	of	ADP
ma-287	764	15	fundamental	fundamental	ADJ
ma-287	764	16	research	research	PROPN
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ma-287	764	21	]	]	X
ma-287	764	22	h.j	h.j	PROPN
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ma-287	764	25	,	,	PUNCT
ma-287	764	26	the	the	DET
ma-287	764	27	equivalence	equivalence	NOUN
ma-287	764	28	of	of	ADP
ma-287	764	29	definitions	definition	NOUN
ma-287	764	30	of	of	ADP
ma-287	764	31	the	the	DET
ma-287	764	32	natural	natural	ADJ
ma-287	764	33	logarithm	logarithm	NOUN
ma-287	764	34	function	function	NOUN
ma-287	764	35	,	,	PUNCT
ma-287	764	36	coll	coll	PROPN
ma-287	764	37	.	.	PUNCT
ma-287	764	38	math	math	PROPN
ma-287	764	39	.	.	PUNCT
ma-287	765	1	j.	j.	PROPN
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ma-287	765	3	(	(	PUNCT
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ma-287	765	5	)	)	PUNCT
ma-287	765	6	,	,	PUNCT
ma-287	765	7	1	1	NUM
ma-287	765	8	-	-	SYM
ma-287	765	9	7	7	NUM
ma-287	765	10	.	.	PUNCT
ma-287	765	11	https	https	NOUN
ma-287	765	12	:	:	PUNCT
ma-287	765	13	//doi.org/10.1080/07468342.2022.2039553.[14	//doi.org/10.1080/07468342.2022.2039553.[14	PROPN
ma-287	765	14	]	]	PUNCT
ma-287	765	15	w.	w.	PROPN
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ma-287	765	17	,	,	PUNCT
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ma-287	765	19	and	and	CCONJ
ma-287	765	20	complex	complex	ADJ
ma-287	765	21	analysis	analysis	NOUN
ma-287	765	22	,	,	PUNCT
ma-287	765	23	mcgraw	mcgraw	PROPN
ma-287	765	24	-	-	PUNCT
ma-287	765	25	hill	hill	PROPN
ma-287	765	26	,	,	PUNCT
ma-287	765	27	new	new	PROPN
ma-287	765	28	york	york	PROPN
ma-287	765	29	,	,	PUNCT
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ma-287	765	31	]	]	X
ma-287	765	32	l.	l.	PROPN
ma-287	765	33	seidel	seidel	PROPN
ma-287	765	34	,	,	PUNCT
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ma-287	765	36	eine	eine	PROPN
ma-287	765	37	darstellung	darstellung	PROPN
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ma-287	765	39	kreisbogens	kreisbogens	PROPN
ma-287	765	40	,	,	PUNCT
ma-287	765	41	des	des	PROPN
ma-287	765	42	logarithmus	logarithmus	PROPN
ma-287	765	43	und	und	PROPN
ma-287	765	44	des	des	PROPN
ma-287	765	45	elliptischen	elliptischen	ADJ
ma-287	765	46	integrales	integrale	NOUN
ma-287	765	47	erster	erster	VERB
ma-287	765	48	art	art	NOUN
ma-287	765	49	durchunendliche	durchunendliche	NOUN
ma-287	765	50	producte	producte	NOUN
ma-287	765	51	,	,	PUNCT
ma-287	765	52	j.	j.	PROPN
ma-287	765	53	reine	reine	PROPN
ma-287	765	54	angew	angew	PROPN
ma-287	765	55	.	.	PUNCT
ma-287	766	1	math	math	NOUN
ma-287	766	2	.	.	PUNCT
ma-287	767	1	73	73	NUM
ma-287	767	2	(	(	PUNCT
ma-287	767	3	1871	1871	NUM
ma-287	767	4	)	)	PUNCT
ma-287	767	5	,	,	PUNCT
ma-287	767	6	273	273	NUM
ma-287	767	7	-	-	SYM
ma-287	767	8	277.[16	277.[16	NUM
ma-287	767	9	]	]	X
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ma-287	767	15	,	,	PUNCT
ma-287	767	16	k.	k.	PROPN
ma-287	767	17	namboothiri	namboothiri	PROPN
ma-287	767	18	,	,	PUNCT
ma-287	767	19	on	on	ADP
ma-287	767	20	some	some	DET
ma-287	767	21	logarithmic	logarithmic	ADJ
ma-287	767	22	inequalities	inequality	NOUN
ma-287	767	23	,	,	PUNCT
ma-287	767	24	adv	adv	PROPN
ma-287	767	25	.	.	PUNCT
ma-287	767	26	math	math	PROPN
ma-287	767	27	.	.	PUNCT
ma-287	767	28	:	:	PUNCT
ma-287	768	1	sci	sci	PROPN
ma-287	768	2	.	.	PUNCT
ma-287	768	3	j.	j.	PROPN
ma-287	768	4	10	10	NUM
ma-287	768	5	(	(	PUNCT
ma-287	768	6	2021	2021	NUM
ma-287	768	7	)	)	PUNCT
ma-287	768	8	,	,	PUNCT
ma-287	768	9	2483	2483	NUM
ma-287	768	10	-	-	SYM
ma-287	768	11	2489	2489	NUM
ma-287	768	12	.	.	PUNCT
ma-287	769	1	https://doi.org/10.37418/amsj.10.5.14.[17	https://doi.org/10.37418/amsj.10.5.14.[17	VERB
ma-287	769	2	]	]	PUNCT
ma-287	769	3	a.	a.	NOUN
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ma-287	769	5	,	,	PUNCT
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ma-287	769	7	and	and	CCONJ
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ma-287	769	9	bounds	bound	NOUN
ma-287	769	10	of	of	ADP
ma-287	769	11	real	real	ADJ
ma-287	769	12	functions	function	NOUN
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ma-287	769	15	arguments	argument	NOUN
ma-287	769	16	,	,	PUNCT
ma-287	769	17	2023	2023	NUM
ma-287	769	18	.	.	PUNCT
ma-287	770	1	preprint	preprint	NOUN
ma-287	770	2	arxiv	arxiv	PROPN
ma-287	770	3	,	,	PUNCT
ma-287	770	4	1	1	NUM
ma-287	770	5	-	-	SYM
ma-287	770	6	7	7	NUM
ma-287	770	7	.	.	PUNCT
ma-287	771	1	https://doi.org/10.48550/arxiv.2309.02479	https://doi.org/10.48550/arxiv.2309.02479	PROPN
ma-287	771	2	.	.	PUNCT
ma-287	772	1	https://doi.org/10.28924/ada/ma.5.12	https://doi.org/10.28924/ada/ma.5.12	PROPN
ma-287	772	2	https://doi.org/10.1007/978-1-4612-0879-2	https://doi.org/10.1007/978-1-4612-0879-2	PROPN
ma-287	772	3	https://doi.org/10.1007/978-1-4612-0879-2	https://doi.org/10.1007/978-1-4612-0879-2	PROPN
ma-287	772	4	https://doi.org/10.4169/math.mag.90.5.353	https://doi.org/10.4169/math.mag.90.5.353	PROPN
ma-287	772	5	https://doi.org/10.1007/s11139-017-9961-y	https://doi.org/10.1007/s11139-017-9961-y	NOUN
ma-287	772	6	https://doi.org/10.1080/00029890.1987.12000678	https://doi.org/10.1080/00029890.1987.12000678	VERB
ma-287	772	7	https://doi.org/10.1080/00029890.1987.12000678	https://doi.org/10.1080/00029890.1987.12000678	PUNCT
ma-287	773	1	https://doi.org/10.21608/ejmaa.2020.312813	https://doi.org/10.21608/ejmaa.2020.312813	PROPN
ma-287	773	2	https://doi.org/10.1080/00029890.2019.1605799	https://doi.org/10.1080/00029890.2019.1605799	PROPN
ma-287	773	3	https://doi.org/10.1080/00029890.2019.1605799	https://doi.org/10.1080/00029890.2019.1605799	PROPN
ma-287	773	4	https://doi.org/10.21608/ejmaa.2020.312859	https://doi.org/10.21608/ejmaa.2020.312859	PROPN
ma-287	773	5	https://doi.org/10.2307/3615890	https://doi.org/10.2307/3615890	ADJ
ma-287	773	6	https://doi.org/10.1080/07468342.2022.2039553	https://doi.org/10.1080/07468342.2022.2039553	NOUN
ma-287	773	7	https://doi.org/10.1080/07468342.2022.2039553	https://doi.org/10.1080/07468342.2022.2039553	NOUN
ma-287	774	1	https://doi.org/10.37418/amsj.10.5.14	https://doi.org/10.37418/amsj.10.5.14	PRON
ma-287	774	2	https://doi.org/10.48550/arxiv.2309.02479	https://doi.org/10.48550/arxiv.2309.02479	PROPN
ma-287	774	3	1	1	NUM
ma-287	774	4	.	.	PUNCT
ma-287	775	1	introduction	introduction	NOUN
ma-287	775	2	2	2	NUM
ma-287	775	3	.	.	PUNCT
ma-287	775	4	results	result	VERB
ma-287	775	5	2.1	2.1	NUM
ma-287	775	6	.	.	PUNCT
ma-287	776	1	a	a	DET
ma-287	776	2	general	general	ADJ
ma-287	776	3	result	result	NOUN
ma-287	776	4	2.2	2.2	NUM
ma-287	776	5	.	.	PUNCT
ma-287	776	6	specific	specific	ADJ
ma-287	776	7	results	result	NOUN
ma-287	776	8	3	3	NUM
ma-287	776	9	.	.	PUNCT
ma-287	776	10	applications	application	NOUN
ma-287	776	11	3.1	3.1	NUM
ma-287	776	12	.	.	PUNCT
ma-287	776	13	product	product	NOUN
ma-287	776	14	expansions	expansion	NOUN
ma-287	776	15	3.2	3.2	NUM
ma-287	776	16	.	.	PUNCT
ma-287	776	17	inequalities	inequality	NOUN
ma-287	776	18	4	4	NUM
ma-287	776	19	.	.	PUNCT
ma-287	776	20	conclusion	conclusion	NOUN
ma-287	776	21	references	reference	NOUN