id	sid	tid	token	lemma	pos
cbi-57	1	1	v2i1	v2i1	ADP
cbi-57	1	2	volume	volume	NOUN
cbi-57	1	3	2	2	NUM
cbi-57	1	4	,	,	PUNCT
cbi-57	1	5	issue	issue	NOUN
cbi-57	1	6	1	1	NUM
cbi-57	1	7	hyperreal	hyperreal	ADJ
cbi-57	1	8	numbers	number	NOUN
cbi-57	1	9	for	for	ADP
cbi-57	1	10	infinite	infinite	ADJ
cbi-57	1	11	divergent	divergent	ADJ
cbi-57	1	12	series	series	NOUN
cbi-57	1	13	jonathan	jonathan	PROPN
cbi-57	1	14	bartlett	bartlett	PROPN
cbi-57	1	15	,	,	PUNCT
cbi-57	1	16	logan	logan	PROPN
cbi-57	1	17	gaastra	gaastra	PROPN
cbi-57	1	18	,	,	PUNCT
cbi-57	1	19	and	and	CCONJ
cbi-57	1	20	david	david	PROPN
cbi-57	1	21	nemati	nemati	PROPN
cbi-57	1	22	doi	doi	PROPN
cbi-57	1	23	:	:	PUNCT
cbi-57	1	24	10.33014	10.33014	NUM
cbi-57	1	25	/	/	SYM
cbi-57	1	26	issn.2640	issn.2640	VERB
cbi-57	1	27	-	-	PUNCT
cbi-57	1	28	5652.2.1.bartlett	5652.2.1.bartlett	NUM
cbi-57	1	29	-	-	PUNCT
cbi-57	1	30	et	et	NOUN
cbi-57	1	31	-	-	PUNCT
cbi-57	1	32	al.1	al.1	PROPN
cbi-57	1	33	abstract	abstract	NOUN
cbi-57	1	34	treating	treat	VERB
cbi-57	1	35	divergent	divergent	ADJ
cbi-57	1	36	series	series	NOUN
cbi-57	1	37	properly	properly	ADV
cbi-57	1	38	has	have	AUX
cbi-57	1	39	been	be	AUX
cbi-57	1	40	an	an	DET
cbi-57	1	41	ongoing	ongoing	ADJ
cbi-57	1	42	issue	issue	NOUN
cbi-57	1	43	in	in	ADP
cbi-57	1	44	mathematics	mathematic	NOUN
cbi-57	1	45	.	.	PUNCT
cbi-57	2	1	however	however	ADV
cbi-57	2	2	,	,	PUNCT
cbi-57	2	3	many	many	ADJ
cbi-57	2	4	of	of	ADP
cbi-57	2	5	the	the	DET
cbi-57	2	6	problems	problem	NOUN
cbi-57	2	7	in	in	ADP
cbi-57	2	8	divergent	divergent	ADJ
cbi-57	2	9	series	series	NOUN
cbi-57	2	10	stem	stem	VERB
cbi-57	2	11	from	from	ADP
cbi-57	2	12	the	the	DET
cbi-57	2	13	fact	fact	NOUN
cbi-57	2	14	that	that	SCONJ
cbi-57	2	15	divergent	divergent	ADJ
cbi-57	2	16	series	series	NOUN
cbi-57	2	17	were	be	AUX
cbi-57	2	18	discovered	discover	VERB
cbi-57	2	19	prior	prior	ADV
cbi-57	2	20	to	to	ADP
cbi-57	2	21	having	have	VERB
cbi-57	2	22	a	a	DET
cbi-57	2	23	number	number	NOUN
cbi-57	2	24	system	system	NOUN
cbi-57	2	25	which	which	PRON
cbi-57	2	26	could	could	AUX
cbi-57	2	27	handle	handle	VERB
cbi-57	2	28	them	they	PRON
cbi-57	2	29	.	.	PUNCT
cbi-57	3	1	the	the	DET
cbi-57	3	2	infinities	infinity	NOUN
cbi-57	3	3	that	that	PRON
cbi-57	3	4	resulted	result	VERB
cbi-57	3	5	from	from	ADP
cbi-57	3	6	divergent	divergent	ADJ
cbi-57	3	7	series	series	NOUN
cbi-57	3	8	led	lead	VERB
cbi-57	3	9	to	to	ADP
cbi-57	3	10	contradictions	contradiction	NOUN
cbi-57	3	11	within	within	ADP
cbi-57	3	12	the	the	DET
cbi-57	3	13	real	real	ADJ
cbi-57	3	14	number	number	NOUN
cbi-57	3	15	system	system	NOUN
cbi-57	3	16	,	,	PUNCT
cbi-57	3	17	but	but	CCONJ
cbi-57	3	18	these	these	DET
cbi-57	3	19	contradictions	contradiction	NOUN
cbi-57	3	20	are	be	AUX
cbi-57	3	21	largely	largely	ADV
cbi-57	3	22	alleviated	alleviate	VERB
cbi-57	3	23	with	with	ADP
cbi-57	3	24	the	the	DET
cbi-57	3	25	hyperreal	hyperreal	ADJ
cbi-57	3	26	number	number	NOUN
cbi-57	3	27	system	system	NOUN
cbi-57	3	28	.	.	PUNCT
cbi-57	4	1	hyperreal	hyperreal	ADJ
cbi-57	4	2	numbers	number	NOUN
cbi-57	4	3	provide	provide	VERB
cbi-57	4	4	a	a	DET
cbi-57	4	5	framework	framework	NOUN
cbi-57	4	6	for	for	ADP
cbi-57	4	7	dealing	deal	VERB
cbi-57	4	8	with	with	ADP
cbi-57	4	9	divergent	divergent	ADJ
cbi-57	4	10	series	series	NOUN
cbi-57	4	11	in	in	ADP
cbi-57	4	12	a	a	DET
cbi-57	4	13	more	more	ADV
cbi-57	4	14	comprehensive	comprehensive	ADJ
cbi-57	4	15	and	and	CCONJ
cbi-57	4	16	tractable	tractable	ADJ
cbi-57	4	17	way	way	NOUN
cbi-57	4	18	.	.	PUNCT
cbi-57	5	1	1	1	NUM
cbi-57	5	2	the	the	DET
cbi-57	5	3	problem	problem	NOUN
cbi-57	5	4	of	of	ADP
cbi-57	5	5	infinite	infinite	ADJ
cbi-57	5	6	series	series	NOUN
cbi-57	5	7	historically	historically	ADV
cbi-57	5	8	,	,	PUNCT
cbi-57	5	9	infinities	infinity	NOUN
cbi-57	5	10	have	have	AUX
cbi-57	5	11	led	lead	VERB
cbi-57	5	12	to	to	ADP
cbi-57	5	13	many	many	ADJ
cbi-57	5	14	problems	problem	NOUN
cbi-57	5	15	in	in	ADP
cbi-57	5	16	mathematics	mathematic	NOUN
cbi-57	5	17	.	.	PUNCT
cbi-57	6	1	infinities	infinity	NOUN
cbi-57	6	2	,	,	PUNCT
cbi-57	6	3	when	when	SCONJ
cbi-57	6	4	not	not	PART
cbi-57	6	5	handled	handle	VERB
cbi-57	6	6	carefully	carefully	ADV
cbi-57	6	7	,	,	PUNCT
cbi-57	6	8	easily	easily	ADV
cbi-57	6	9	lead	lead	VERB
cbi-57	6	10	to	to	ADP
cbi-57	6	11	contradictions	contradiction	NOUN
cbi-57	6	12	and	and	CCONJ
cbi-57	6	13	indeterminacies	indeterminacy	NOUN
cbi-57	6	14	.	.	PUNCT
cbi-57	7	1	therefore	therefore	ADV
cbi-57	7	2	,	,	PUNCT
cbi-57	7	3	caution	caution	NOUN
cbi-57	7	4	has	have	AUX
cbi-57	7	5	always	always	ADV
cbi-57	7	6	been	be	AUX
cbi-57	7	7	urged	urge	VERB
cbi-57	7	8	when	when	SCONJ
cbi-57	7	9	dealing	deal	VERB
cbi-57	7	10	with	with	ADP
cbi-57	7	11	infinite	infinite	ADJ
cbi-57	7	12	series	series	NOUN
cbi-57	7	13	.	.	PUNCT
cbi-57	8	1	this	this	PRON
cbi-57	8	2	is	be	AUX
cbi-57	8	3	especially	especially	ADV
cbi-57	8	4	true	true	ADJ
cbi-57	8	5	with	with	ADP
cbi-57	8	6	divergent	divergent	ADJ
cbi-57	8	7	infinite	infinite	ADJ
cbi-57	8	8	series	series	NOUN
cbi-57	8	9	.	.	PUNCT
cbi-57	9	1	convergent	convergent	NOUN
cbi-57	9	2	infinite	infinite	PROPN
cbi-57	9	3	series	series	NOUN
cbi-57	9	4	generally	generally	ADV
cbi-57	9	5	behave	behave	VERB
cbi-57	9	6	unproblematically	unproblematically	ADV
cbi-57	9	7	similar	similar	ADJ
cbi-57	9	8	to	to	ADP
cbi-57	9	9	the	the	DET
cbi-57	9	10	value	value	NOUN
cbi-57	9	11	that	that	PRON
cbi-57	9	12	they	they	PRON
cbi-57	9	13	converge	converge	VERB
cbi-57	9	14	to	to	ADP
cbi-57	9	15	.	.	PUNCT
cbi-57	10	1	given	give	VERB
cbi-57	10	2	a	a	DET
cbi-57	10	3	series	series	NOUN
cbi-57	10	4	that	that	PRON
cbi-57	10	5	converges	converge	VERB
cbi-57	10	6	to	to	ADP
cbi-57	10	7	2	2	NUM
cbi-57	10	8	and	and	CCONJ
cbi-57	10	9	another	another	DET
cbi-57	10	10	series	series	NOUN
cbi-57	10	11	that	that	PRON
cbi-57	10	12	converges	converge	VERB
cbi-57	10	13	to	to	ADP
cbi-57	10	14	3	3	NUM
cbi-57	10	15	then	then	ADV
cbi-57	10	16	the	the	DET
cbi-57	10	17	sum	sum	NOUN
cbi-57	10	18	of	of	ADP
cbi-57	10	19	the	the	DET
cbi-57	10	20	values	value	NOUN
cbi-57	10	21	of	of	ADP
cbi-57	10	22	the	the	DET
cbi-57	10	23	series	series	NOUN
cbi-57	10	24	will	will	AUX
cbi-57	10	25	be	be	AUX
cbi-57	10	26	5	5	NUM
cbi-57	10	27	and	and	CCONJ
cbi-57	10	28	their	their	PRON
cbi-57	10	29	product	product	NOUN
cbi-57	10	30	will	will	AUX
cbi-57	10	31	be	be	AUX
cbi-57	10	32	6	6	NUM
cbi-57	10	33	.	.	PUNCT
cbi-57	11	1	therefore	therefore	ADV
cbi-57	11	2	,	,	PUNCT
cbi-57	11	3	the	the	DET
cbi-57	11	4	nature	nature	NOUN
cbi-57	11	5	of	of	ADP
cbi-57	11	6	these	these	DET
cbi-57	11	7	series	series	NOUN
cbi-57	11	8	can	can	AUX
cbi-57	11	9	be	be	AUX
cbi-57	11	10	summarized	summarize	VERB
cbi-57	11	11	into	into	ADP
cbi-57	11	12	a	a	DET
cbi-57	11	13	single	single	ADJ
cbi-57	11	14	number	number	NOUN
cbi-57	11	15	.	.	PUNCT
cbi-57	12	1	with	with	ADP
cbi-57	12	2	divergent	divergent	ADJ
cbi-57	12	3	series	series	NOUN
cbi-57	12	4	,	,	PUNCT
cbi-57	12	5	this	this	PRON
cbi-57	12	6	is	be	AUX
cbi-57	12	7	not	not	PART
cbi-57	12	8	so	so	ADV
cbi-57	12	9	straightforward	straightforward	ADJ
cbi-57	12	10	.	.	PUNCT
cbi-57	13	1	a	a	DET
cbi-57	13	2	lack	lack	NOUN
cbi-57	13	3	of	of	ADP
cbi-57	13	4	agreement	agreement	NOUN
cbi-57	13	5	on	on	ADP
cbi-57	13	6	the	the	DET
cbi-57	13	7	rules	rule	NOUN
cbi-57	13	8	for	for	ADP
cbi-57	13	9	handling	handle	VERB
cbi-57	13	10	infinities	infinity	NOUN
cbi-57	13	11	had	have	AUX
cbi-57	13	12	led	lead	VERB
cbi-57	13	13	to	to	ADP
cbi-57	13	14	numerous	numerous	ADJ
cbi-57	13	15	problems	problem	NOUN
cbi-57	13	16	with	with	ADP
cbi-57	13	17	handling	handle	VERB
cbi-57	13	18	divergent	divergent	ADJ
cbi-57	13	19	series	series	NOUN
cbi-57	13	20	.	.	PUNCT
cbi-57	14	1	if	if	SCONJ
cbi-57	14	2	a	a	DET
cbi-57	14	3	series	series	NOUN
cbi-57	14	4	diverges	diverge	VERB
cbi-57	14	5	to	to	ADP
cbi-57	14	6	infinity	infinity	NOUN
cbi-57	14	7	,	,	PUNCT
cbi-57	14	8	is	be	AUX
cbi-57	14	9	it	it	PRON
cbi-57	14	10	greater	great	ADJ
cbi-57	14	11	than	than	ADP
cbi-57	14	12	or	or	CCONJ
cbi-57	14	13	equal	equal	ADJ
cbi-57	14	14	to	to	ADP
cbi-57	14	15	some	some	DET
cbi-57	14	16	other	other	ADJ
cbi-57	14	17	series	series	NOUN
cbi-57	14	18	that	that	PRON
cbi-57	14	19	diverges	diverge	VERB
cbi-57	14	20	to	to	ADP
cbi-57	14	21	infinity	infinity	NOUN
cbi-57	14	22	?	?	PUNCT
cbi-57	15	1	can	can	AUX
cbi-57	15	2	the	the	DET
cbi-57	15	3	terms	term	NOUN
cbi-57	15	4	of	of	ADP
cbi-57	15	5	the	the	DET
cbi-57	15	6	series	series	NOUN
cbi-57	15	7	be	be	AUX
cbi-57	15	8	rearranged	rearrange	VERB
cbi-57	15	9	?	?	PUNCT
cbi-57	16	1	can	can	AUX
cbi-57	16	2	their	their	PRON
cbi-57	16	3	spacing	spacing	NOUN
cbi-57	16	4	be	be	AUX
cbi-57	16	5	modified	modify	VERB
cbi-57	16	6	?	?	PUNCT
cbi-57	17	1	is	be	AUX
cbi-57	17	2	1	1	NUM
cbi-57	17	3	+	+	CCONJ
cbi-57	17	4	1	1	NUM
cbi-57	17	5	+	+	NUM
cbi-57	17	6	1	1	NUM
cbi-57	17	7	+	+	NUM
cbi-57	17	8	.	.	PUNCT
cbi-57	17	9	.	.	PUNCT
cbi-57	17	10	.	.	PUNCT
cbi-57	18	1	equivalent	equivalent	ADJ
cbi-57	18	2	to	to	ADP
cbi-57	18	3	1	1	NUM
cbi-57	18	4	+	+	CCONJ
cbi-57	18	5	0	0	NUM
cbi-57	19	1	+	+	CCONJ
cbi-57	19	2	1	1	NUM
cbi-57	19	3	+	+	SYM
cbi-57	19	4	0	0	NUM
cbi-57	20	1	+	+	CCONJ
cbi-57	20	2	1	1	NUM
cbi-57	20	3	+	+	NUM
cbi-57	20	4	0	0	NUM
cbi-57	21	1	+	+	CCONJ
cbi-57	21	2	.	.	PUNCT
cbi-57	21	3	.	.	PUNCT
cbi-57	21	4	.	.	PUNCT
cbi-57	21	5	?	?	PUNCT
cbi-57	22	1	lack	lack	NOUN
cbi-57	22	2	of	of	ADP
cbi-57	22	3	answers	answer	NOUN
cbi-57	22	4	to	to	ADP
cbi-57	22	5	questions	question	NOUN
cbi-57	22	6	like	like	ADP
cbi-57	22	7	this	this	PRON
cbi-57	22	8	have	have	AUX
cbi-57	22	9	stifled	stifle	VERB
cbi-57	22	10	work	work	NOUN
cbi-57	22	11	in	in	ADP
cbi-57	22	12	divergent	divergent	ADJ
cbi-57	22	13	series	series	NOUN
cbi-57	22	14	,	,	PUNCT
cbi-57	22	15	and	and	CCONJ
cbi-57	22	16	have	have	AUX
cbi-57	22	17	caused	cause	VERB
cbi-57	22	18	many	many	ADJ
cbi-57	22	19	mathematicians	mathematician	NOUN
cbi-57	22	20	to	to	PART
cbi-57	22	21	think	think	VERB
cbi-57	22	22	of	of	ADP
cbi-57	22	23	divergent	divergent	ADJ
cbi-57	22	24	series	series	NOUN
cbi-57	22	25	as	as	ADP
cbi-57	22	26	invalid	invalid	ADJ
cbi-57	22	27	entities	entity	NOUN
cbi-57	22	28	to	to	PART
cbi-57	22	29	work	work	VERB
cbi-57	22	30	with	with	ADP
cbi-57	22	31	rigorously	rigorously	ADV
cbi-57	22	32	.	.	PUNCT
cbi-57	23	1	2	2	NUM
cbi-57	23	2	working	work	VERB
cbi-57	23	3	with	with	ADP
cbi-57	23	4	infinities	infinity	NOUN
cbi-57	23	5	many	many	ADJ
cbi-57	23	6	paradoxes	paradox	NOUN
cbi-57	23	7	exist	exist	VERB
cbi-57	23	8	with	with	ADP
cbi-57	23	9	infinities	infinity	NOUN
cbi-57	23	10	.	.	PUNCT
cbi-57	24	1	for	for	ADP
cbi-57	24	2	instance	instance	NOUN
cbi-57	24	3	,	,	PUNCT
cbi-57	24	4	are	be	AUX
cbi-57	24	5	there	there	PRON
cbi-57	24	6	the	the	DET
cbi-57	24	7	same	same	ADJ
cbi-57	24	8	number	number	NOUN
cbi-57	24	9	of	of	ADP
cbi-57	24	10	positive	positive	ADJ
cbi-57	24	11	even	even	ADV
cbi-57	24	12	integers	integer	NOUN
cbi-57	24	13	as	as	ADP
cbi-57	24	14	positive	positive	ADJ
cbi-57	24	15	integers	integer	NOUN
cbi-57	24	16	?	?	PUNCT
cbi-57	25	1	there	there	PRON
cbi-57	25	2	are	be	VERB
cbi-57	25	3	an	an	DET
cbi-57	25	4	infinity	infinity	NOUN
cbi-57	25	5	of	of	ADP
cbi-57	25	6	them	they	PRON
cbi-57	25	7	,	,	PUNCT
cbi-57	25	8	but	but	CCONJ
cbi-57	25	9	does	do	AUX
cbi-57	25	10	that	that	PRON
cbi-57	25	11	make	make	VERB
cbi-57	25	12	them	they	PRON
cbi-57	25	13	the	the	DET
cbi-57	25	14	same	same	ADJ
cbi-57	25	15	?	?	PUNCT
cbi-57	26	1	it	it	PRON
cbi-57	26	2	seems	seem	VERB
cbi-57	26	3	pretty	pretty	ADV
cbi-57	26	4	obvious	obvious	ADJ
cbi-57	26	5	that	that	SCONJ
cbi-57	26	6	,	,	PUNCT
cbi-57	26	7	on	on	ADP
cbi-57	26	8	a	a	DET
cbi-57	26	9	number	number	NOUN
cbi-57	26	10	line	line	NOUN
cbi-57	26	11	,	,	PUNCT
cbi-57	26	12	positive	positive	ADJ
cbi-57	26	13	integers	integer	NOUN
cbi-57	26	14	occur	occur	VERB
cbi-57	26	15	twice	twice	ADV
cbi-57	26	16	as	as	ADV
cbi-57	26	17	often	often	ADV
cbi-57	26	18	.	.	PUNCT
cbi-57	27	1	however	however	ADV
cbi-57	27	2	,	,	PUNCT
cbi-57	27	3	there	there	PRON
cbi-57	27	4	are	be	VERB
cbi-57	27	5	an	an	DET
cbi-57	27	6	infinite	infinite	ADJ
cbi-57	27	7	amount	amount	NOUN
cbi-57	27	8	of	of	ADP
cbi-57	27	9	both	both	PRON
cbi-57	27	10	.	.	PUNCT
cbi-57	28	1	cantor	cantor	PROPN
cbi-57	28	2	’s	’s	PART
cbi-57	28	3	solution	solution	NOUN
cbi-57	28	4	to	to	ADP
cbi-57	28	5	this	this	DET
cbi-57	28	6	problem	problem	NOUN
cbi-57	28	7	is	be	AUX
cbi-57	28	8	to	to	PART
cbi-57	28	9	separate	separate	VERB
cbi-57	28	10	out	out	ADP
cbi-57	28	11	the	the	DET
cbi-57	28	12	final	final	ADJ
cbi-57	28	13	quantity	quantity	NOUN
cbi-57	28	14	of	of	ADP
cbi-57	28	15	a	a	DET
cbi-57	28	16	set	set	NOUN
cbi-57	28	17	(	(	PUNCT
cbi-57	28	18	the	the	DET
cbi-57	28	19	cardinality	cardinality	NOUN
cbi-57	28	20	)	)	PUNCT
cbi-57	28	21	from	from	ADP
cbi-57	28	22	the	the	DET
cbi-57	28	23	arrangment	arrangment	NOUN
cbi-57	28	24	of	of	ADP
cbi-57	28	25	a	a	DET
cbi-57	28	26	set	set	NOUN
cbi-57	28	27	(	(	PUNCT
cbi-57	28	28	its	its	PRON
cbi-57	28	29	ordinality	ordinality	NOUN
cbi-57	28	30	)	)	PUNCT
cbi-57	28	31	.	.	PUNCT
cbi-57	29	1	the	the	DET
cbi-57	29	2	cardinal	cardinal	ADJ
cbi-57	29	3	numbers	number	NOUN
cbi-57	29	4	do	do	AUX
cbi-57	29	5	not	not	PART
cbi-57	29	6	behave	behave	VERB
cbi-57	29	7	in	in	ADP
cbi-57	29	8	any	any	DET
cbi-57	29	9	way	way	NOUN
cbi-57	29	10	similar	similar	ADJ
cbi-57	29	11	to	to	ADP
cbi-57	29	12	real	real	ADJ
cbi-57	29	13	numbers	number	NOUN
cbi-57	29	14	.	.	PUNCT
cbi-57	30	1	the	the	DET
cbi-57	30	2	ordinals	ordinal	NOUN
cbi-57	30	3	,	,	PUNCT
cbi-57	30	4	on	on	ADP
cbi-57	30	5	the	the	DET
cbi-57	30	6	other	other	ADJ
cbi-57	30	7	hand	hand	NOUN
cbi-57	30	8	,	,	PUNCT
cbi-57	30	9	behave	behave	VERB
cbi-57	30	10	in	in	ADP
cbi-57	30	11	many	many	ADJ
cbi-57	30	12	ways	way	NOUN
cbi-57	30	13	similar	similar	ADJ
cbi-57	30	14	to	to	ADP
cbi-57	30	15	real	real	ADJ
cbi-57	30	16	numbers	number	NOUN
cbi-57	30	17	.	.	PUNCT
cbi-57	31	1	however	however	ADV
cbi-57	31	2	,	,	PUNCT
cbi-57	31	3	cantor	cantor	PROPN
cbi-57	31	4	’s	’s	PART
cbi-57	31	5	own	own	ADJ
cbi-57	31	6	system	system	NOUN
cbi-57	31	7	for	for	ADP
cbi-57	31	8	ordinal	ordinal	ADJ
cbi-57	31	9	arithmetic	arithmetic	NOUN
cbi-57	31	10	is	be	AUX
cbi-57	31	11	difficult	difficult	ADJ
cbi-57	31	12	to	to	PART
cbi-57	31	13	use	use	VERB
cbi-57	31	14	,	,	PUNCT
cbi-57	31	15	and	and	CCONJ
cbi-57	31	16	does	do	AUX
cbi-57	31	17	n’t	not	PART
cbi-57	31	18	translate	translate	VERB
cbi-57	31	19	well	well	ADV
cbi-57	31	20	between	between	ADP
cbi-57	31	21	transfinite	transfinite	ADJ
cbi-57	31	22	and	and	CCONJ
cbi-57	31	23	regular	regular	ADJ
cbi-57	31	24	real	real	ADJ
cbi-57	31	25	arithmetic	arithmetic	NOUN
cbi-57	31	26	.	.	PUNCT
cbi-57	32	1	the	the	DET
cbi-57	32	2	hyperreal	hyperreal	ADJ
cbi-57	32	3	number	number	NOUN
cbi-57	32	4	line	line	NOUN
cbi-57	32	5	has	have	VERB
cbi-57	32	6	many	many	ADJ
cbi-57	32	7	similarities	similarity	NOUN
cbi-57	32	8	to	to	ADP
cbi-57	32	9	cantor	cantor	PROPN
cbi-57	32	10	’s	’s	PART
cbi-57	32	11	ordinals	ordinal	NOUN
cbi-57	32	12	,	,	PUNCT
cbi-57	32	13	operating	operate	VERB
cbi-57	32	14	essentially	essentially	ADV
cbi-57	32	15	at	at	ADP
cbi-57	32	16	the	the	DET
cbi-57	32	17	level	level	NOUN
cbi-57	32	18	of	of	ADP
cbi-57	32	19	“	"	PUNCT
cbi-57	32	20	ordinal	ordinal	ADJ
cbi-57	32	21	”	"	PUNCT
cbi-57	32	22	in	in	ADP
cbi-57	32	23	cantor	cantor	PROPN
cbi-57	32	24	’s	’s	PART
cbi-57	32	25	system	system	NOUN
cbi-57	32	26	.	.	PUNCT
cbi-57	33	1	however	however	ADV
cbi-57	33	2	,	,	PUNCT
cbi-57	33	3	the	the	DET
cbi-57	33	4	hyperreal	hyperreal	ADJ
cbi-57	33	5	number	number	NOUN
cbi-57	33	6	line	line	NOUN
cbi-57	33	7	o!ers	o!er	VERB
cbi-57	33	8	a	a	DET
cbi-57	33	9	way	way	NOUN
cbi-57	33	10	to	to	PART
cbi-57	33	11	do	do	VERB
cbi-57	33	12	arithmetic	arithmetic	NOUN
cbi-57	33	13	with	with	ADP
cbi-57	33	14	infinities	infinity	NOUN
cbi-57	33	15	in	in	ADP
cbi-57	33	16	a	a	DET
cbi-57	33	17	way	way	NOUN
cbi-57	33	18	that	that	PRON
cbi-57	33	19	very	very	ADV
cbi-57	33	20	closely	closely	ADV
cbi-57	33	21	matches	match	VERB
cbi-57	33	22	real	real	ADJ
cbi-57	33	23	arithmetic	arithmetic	ADJ
cbi-57	33	24	through	through	ADP
cbi-57	33	25	the	the	DET
cbi-57	33	26	use	use	NOUN
cbi-57	33	27	of	of	ADP
cbi-57	33	28	the	the	DET
cbi-57	33	29	transfer	transfer	NOUN
cbi-57	33	30	principle	principle	NOUN
cbi-57	33	31	(	(	PUNCT
cbi-57	33	32	henle	henle	NOUN
cbi-57	33	33	and	and	CCONJ
cbi-57	33	34	kleinberg	kleinberg	PROPN
cbi-57	33	35	,	,	PUNCT
cbi-57	33	36	2003	2003	NUM
cbi-57	33	37	)	)	PUNCT
cbi-57	33	38	.	.	PUNCT
cbi-57	34	1	the	the	DET
cbi-57	34	2	transfer	transfer	NOUN
cbi-57	34	3	principle	principle	NOUN
cbi-57	34	4	states	state	NOUN
cbi-57	34	5	that	that	SCONJ
cbi-57	34	6	any	any	DET
cbi-57	34	7	first	first	ADJ
cbi-57	34	8	-	-	PUNCT
cbi-57	34	9	order	order	NOUN
cbi-57	34	10	proposition	proposition	NOUN
cbi-57	34	11	that	that	PRON
cbi-57	34	12	is	be	AUX
cbi-57	34	13	true	true	ADJ
cbi-57	34	14	for	for	ADP
cbi-57	34	15	the	the	DET
cbi-57	34	16	reals	real	NOUN
cbi-57	34	17	is	be	AUX
cbi-57	34	18	also	also	ADV
cbi-57	34	19	true	true	ADJ
cbi-57	34	20	for	for	ADP
cbi-57	34	21	the	the	DET
cbi-57	34	22	hyperreals	hyperreal	NOUN
cbi-57	34	23	.	.	PUNCT
cbi-57	35	1	this	this	PRON
cbi-57	35	2	means	mean	VERB
cbi-57	35	3	that	that	SCONJ
cbi-57	35	4	the	the	DET
cbi-57	35	5	standard	standard	ADJ
cbi-57	35	6	arithmetic	arithmetic	ADJ
cbi-57	35	7	principles	principle	NOUN
cbi-57	35	8	for	for	ADP
cbi-57	35	9	dealing	deal	VERB
cbi-57	35	10	with	with	ADP
cbi-57	35	11	real	real	ADJ
cbi-57	35	12	numbers	number	NOUN
cbi-57	35	13	will	will	AUX
cbi-57	35	14	apply	apply	VERB
cbi-57	35	15	to	to	ADP
cbi-57	35	16	hyperreal	hyperreal	ADJ
cbi-57	35	17	numbers	number	NOUN
cbi-57	35	18	as	as	ADV
cbi-57	35	19	well	well	ADV
cbi-57	35	20	.	.	PUNCT
cbi-57	36	1	the	the	DET
cbi-57	36	2	hyperreal	hyperreal	ADJ
cbi-57	36	3	number	number	NOUN
cbi-57	36	4	line	line	NOUN
cbi-57	36	5	operates	operate	VERB
cbi-57	36	6	with	with	ADP
cbi-57	36	7	an	an	DET
cbi-57	36	8	infinite	infinite	ADJ
cbi-57	36	9	unit	unit	NOUN
cbi-57	36	10	,	,	PUNCT
cbi-57	36	11	!	!	PUNCT
cbi-57	36	12	,	,	PUNCT
cbi-57	36	13	that	that	PRON
cbi-57	36	14	represents	represent	VERB
cbi-57	36	15	an	an	DET
cbi-57	36	16	order	order	NOUN
cbi-57	36	17	of	of	ADP
cbi-57	36	18	infinity.1	infinity.1	PROPN
cbi-57	36	19	the	the	DET
cbi-57	36	20	way	way	NOUN
cbi-57	36	21	it	it	PRON
cbi-57	36	22	is	be	AUX
cbi-57	36	23	usually	usually	ADV
cbi-57	36	24	handled	handle	VERB
cbi-57	36	25	,	,	PUNCT
cbi-57	36	26	!	!	PUNCT
cbi-57	37	1	is	be	AUX
cbi-57	37	2	n’t	not	PART
cbi-57	37	3	a	a	DET
cbi-57	37	4	specific	specific	ADJ
cbi-57	37	5	number	number	NOUN
cbi-57	37	6	in	in	ADP
cbi-57	37	7	the	the	DET
cbi-57	37	8	typical	typical	ADJ
cbi-57	37	9	sense	sense	NOUN
cbi-57	37	10	,	,	PUNCT
cbi-57	37	11	but	but	CCONJ
cbi-57	37	12	rather	rather	ADV
cbi-57	37	13	more	more	ADJ
cbi-57	37	14	of	of	ADP
cbi-57	37	15	a	a	DET
cbi-57	37	16	benchmark	benchmark	NOUN
cbi-57	37	17	of	of	ADP
cbi-57	37	18	infinity	infinity	NOUN
cbi-57	37	19	.	.	PUNCT
cbi-57	38	1	previous	previous	ADJ
cbi-57	38	2	work	work	NOUN
cbi-57	38	3	has	have	AUX
cbi-57	38	4	shown	show	VERB
cbi-57	38	5	that	that	SCONJ
cbi-57	38	6	hyperreal	hyperreal	ADJ
cbi-57	38	7	numbers	number	NOUN
cbi-57	38	8	could	could	AUX
cbi-57	38	9	be	be	AUX
cbi-57	38	10	a	a	DET
cbi-57	38	11	potential	potential	ADJ
cbi-57	38	12	solution	solution	NOUN
cbi-57	38	13	to	to	ADP
cbi-57	38	14	how	how	SCONJ
cbi-57	38	15	values	value	NOUN
cbi-57	38	16	of	of	ADP
cbi-57	38	17	divergent	divergent	ADJ
cbi-57	38	18	series	series	NOUN
cbi-57	38	19	can	can	AUX
cbi-57	38	20	be	be	AUX
cbi-57	38	21	represented	represent	VERB
cbi-57	38	22	(	(	PUNCT
cbi-57	38	23	gaastra	gaastra	PROPN
cbi-57	38	24	,	,	PUNCT
cbi-57	38	25	2016).2	2016).2	NUM
cbi-57	38	26	the	the	DET
cbi-57	38	27	present	present	ADJ
cbi-57	38	28	paper	paper	NOUN
cbi-57	38	29	will	will	AUX
cbi-57	38	30	build	build	VERB
cbi-57	38	31	on	on	ADP
cbi-57	38	32	this	this	DET
cbi-57	38	33	original	original	ADJ
cbi-57	38	34	idea	idea	NOUN
cbi-57	38	35	and	and	CCONJ
cbi-57	38	36	establish	establish	VERB
cbi-57	38	37	a	a	DET
cbi-57	38	38	system	system	NOUN
cbi-57	38	39	for	for	ADP
cbi-57	38	40	using	use	VERB
cbi-57	38	41	1the	1the	ADJ
cbi-57	38	42	choice	choice	NOUN
cbi-57	38	43	of	of	ADP
cbi-57	38	44	character	character	NOUN
cbi-57	38	45	/	/	SYM
cbi-57	38	46	typography	typography	NOUN
cbi-57	38	47	for	for	ADP
cbi-57	38	48	the	the	DET
cbi-57	38	49	unit	unit	NOUN
cbi-57	38	50	varies	vary	VERB
cbi-57	38	51	with	with	ADP
cbi-57	38	52	the	the	DET
cbi-57	38	53	author	author	NOUN
cbi-57	38	54	.	.	PUNCT
cbi-57	39	1	for	for	ADP
cbi-57	39	2	instance	instance	NOUN
cbi-57	39	3	,	,	PUNCT
cbi-57	39	4	keisler	keisler	PROPN
cbi-57	39	5	uses	use	VERB
cbi-57	39	6	h	h	PROPN
cbi-57	39	7	(	(	PUNCT
cbi-57	39	8	keisler	keisler	NOUN
cbi-57	39	9	,	,	PUNCT
cbi-57	39	10	2012	2012	NUM
cbi-57	39	11	)	)	PUNCT
cbi-57	39	12	.	.	PUNCT
cbi-57	40	1	!	!	PUNCT
cbi-57	40	2	was	be	AUX
cbi-57	40	3	chosen	choose	VERB
cbi-57	40	4	because	because	SCONJ
cbi-57	40	5	of	of	ADP
cbi-57	40	6	its	its	PRON
cbi-57	40	7	historical	historical	ADJ
cbi-57	40	8	connection	connection	NOUN
cbi-57	40	9	with	with	ADP
cbi-57	40	10	ordinal	ordinal	ADJ
cbi-57	40	11	-	-	PUNCT
cbi-57	40	12	type	type	NOUN
cbi-57	40	13	infinities	infinity	NOUN
cbi-57	40	14	.	.	PUNCT
cbi-57	41	1	2other	2other	NUM
cbi-57	41	2	work	work	VERB
cbi-57	41	3	worth	worth	ADJ
cbi-57	41	4	mentioning	mention	VERB
cbi-57	41	5	in	in	ADP
cbi-57	41	6	this	this	DET
cbi-57	41	7	area	area	NOUN
cbi-57	41	8	are	be	AUX
cbi-57	41	9	(	(	PUNCT
cbi-57	41	10	paterson	paterson	NOUN
cbi-57	41	11	,	,	PUNCT
cbi-57	41	12	2018a	2018a	NUM
cbi-57	41	13	)	)	PUNCT
cbi-57	41	14	and	and	CCONJ
cbi-57	41	15	(	(	PUNCT
cbi-57	41	16	paterson	paterson	NOUN
cbi-57	41	17	,	,	PUNCT
cbi-57	41	18	2018b	2018b	NUM
cbi-57	41	19	)	)	PUNCT
cbi-57	41	20	.	.	PUNCT
cbi-57	42	1	in	in	ADP
cbi-57	42	2	the	the	DET
cbi-57	42	3	current	current	ADJ
cbi-57	42	4	work	work	NOUN
cbi-57	42	5	,	,	PUNCT
cbi-57	42	6	we	we	PRON
cbi-57	42	7	will	will	AUX
cbi-57	42	8	use	use	VERB
cbi-57	42	9	a	a	DET
cbi-57	42	10	notation	notation	NOUN
cbi-57	42	11	similar	similar	ADJ
cbi-57	42	12	to	to	ADP
cbi-57	42	13	(	(	PUNCT
cbi-57	42	14	keisler	keisler	NOUN
cbi-57	42	15	,	,	PUNCT
cbi-57	42	16	2012	2012	NUM
cbi-57	42	17	)	)	PUNCT
cbi-57	42	18	to	to	PART
cbi-57	42	19	notate	notate	VERB
cbi-57	42	20	hyperreal	hyperreal	ADJ
cbi-57	42	21	values	value	NOUN
cbi-57	42	22	,	,	PUNCT
cbi-57	42	23	and	and	CCONJ
cbi-57	42	24	show	show	VERB
cbi-57	42	25	how	how	SCONJ
cbi-57	42	26	infinite	infinite	ADJ
cbi-57	42	27	series	series	NOUN
cbi-57	42	28	can	can	AUX
cbi-57	42	29	be	be	AUX
cbi-57	42	30	simplified	simplify	VERB
cbi-57	42	31	to	to	ADP
cbi-57	42	32	them	they	PRON
cbi-57	42	33	.	.	PUNCT
cbi-57	43	1	paterson	paterson	PROPN
cbi-57	43	2	did	do	VERB
cbi-57	43	3	the	the	DET
cbi-57	43	4	opposite	opposite	NOUN
cbi-57	43	5	,	,	PUNCT
cbi-57	43	6	by	by	ADP
cbi-57	43	7	notating	notate	VERB
cbi-57	43	8	hyperreal	hyperreal	ADJ
cbi-57	43	9	values	value	NOUN
cbi-57	43	10	with	with	ADP
cbi-57	43	11	the	the	DET
cbi-57	43	12	infinite	infinite	ADJ
cbi-57	43	13	sum	sum	NOUN
cbi-57	43	14	that	that	PRON
cbi-57	43	15	represents	represent	VERB
cbi-57	43	16	them	they	PRON
cbi-57	43	17	.	.	PUNCT
cbi-57	44	1	8	8	NUM
cbi-57	44	2	hyperreal	hyperreal	ADJ
cbi-57	44	3	numbers	number	NOUN
cbi-57	44	4	for	for	ADP
cbi-57	44	5	infinite	infinite	ADJ
cbi-57	44	6	divergent	divergent	ADJ
cbi-57	44	7	series	series	NOUN
cbi-57	44	8	hyperreal	hyperreal	NOUN
cbi-57	44	9	numbers	number	NOUN
cbi-57	44	10	to	to	PART
cbi-57	44	11	assign	assign	VERB
cbi-57	44	12	values	value	NOUN
cbi-57	44	13	to	to	PART
cbi-57	44	14	infinite	infinite	VERB
cbi-57	44	15	series	series	NOUN
cbi-57	44	16	.	.	PUNCT
cbi-57	45	1	3	3	NUM
cbi-57	45	2	hyperreals	hyperreal	NOUN
cbi-57	45	3	and	and	CCONJ
cbi-57	45	4	partial	partial	ADJ
cbi-57	45	5	sums	sum	VERB
cbi-57	45	6	the	the	DET
cbi-57	45	7	vast	vast	ADJ
cbi-57	45	8	majority	majority	NOUN
cbi-57	45	9	of	of	ADP
cbi-57	45	10	issues	issue	NOUN
cbi-57	45	11	with	with	ADP
cbi-57	45	12	divergent	divergent	ADJ
cbi-57	45	13	series	series	NOUN
cbi-57	45	14	comes	come	VERB
cbi-57	45	15	with	with	ADP
cbi-57	45	16	the	the	DET
cbi-57	45	17	transition	transition	NOUN
cbi-57	45	18	from	from	ADP
cbi-57	45	19	partial	partial	ADJ
cbi-57	45	20	sums	sum	NOUN
cbi-57	45	21	to	to	PART
cbi-57	45	22	infinite	infinite	VERB
cbi-57	45	23	sums	sum	NOUN
cbi-57	45	24	.	.	PUNCT
cbi-57	46	1	as	as	ADV
cbi-57	46	2	long	long	ADV
cbi-57	46	3	as	as	SCONJ
cbi-57	46	4	a	a	DET
cbi-57	46	5	series	series	NOUN
cbi-57	46	6	remains	remain	VERB
cbi-57	46	7	a	a	DET
cbi-57	46	8	partial	partial	ADJ
cbi-57	46	9	sum	sum	NOUN
cbi-57	46	10	,	,	PUNCT
cbi-57	46	11	arithmetic	arithmetic	ADJ
cbi-57	46	12	with	with	ADP
cbi-57	46	13	the	the	DET
cbi-57	46	14	series	series	NOUN
cbi-57	46	15	is	be	AUX
cbi-57	46	16	unproblematic	unproblematic	ADJ
cbi-57	46	17	.	.	PUNCT
cbi-57	47	1	therefore	therefore	ADV
cbi-57	47	2	,	,	PUNCT
cbi-57	47	3	it	it	PRON
cbi-57	47	4	would	would	AUX
cbi-57	47	5	be	be	AUX
cbi-57	47	6	beneficial	beneficial	ADJ
cbi-57	47	7	to	to	PART
cbi-57	47	8	develop	develop	VERB
cbi-57	47	9	a	a	DET
cbi-57	47	10	system	system	NOUN
cbi-57	47	11	which	which	PRON
cbi-57	47	12	matched	match	VERB
cbi-57	47	13	the	the	DET
cbi-57	47	14	partial	partial	ADJ
cbi-57	47	15	sum	sum	NOUN
cbi-57	47	16	behavior	behavior	NOUN
cbi-57	47	17	of	of	ADP
cbi-57	47	18	finite	finite	ADJ
cbi-57	47	19	sums	sum	NOUN
cbi-57	47	20	,	,	PUNCT
cbi-57	47	21	but	but	CCONJ
cbi-57	47	22	allowed	allow	VERB
cbi-57	47	23	the	the	DET
cbi-57	47	24	result	result	NOUN
cbi-57	47	25	to	to	PART
cbi-57	47	26	be	be	AUX
cbi-57	47	27	generalized	generalize	VERB
cbi-57	47	28	to	to	ADP
cbi-57	47	29	infinity	infinity	NOUN
cbi-57	47	30	.	.	PUNCT
cbi-57	48	1	the	the	DET
cbi-57	48	2	value	value	NOUN
cbi-57	48	3	of	of	ADP
cbi-57	48	4	a	a	DET
cbi-57	48	5	partial	partial	ADJ
cbi-57	48	6	sum	sum	NOUN
cbi-57	48	7	of	of	ADP
cbi-57	48	8	an	an	DET
cbi-57	48	9	infinite	infinite	ADJ
cbi-57	48	10	sequence	sequence	NOUN
cbi-57	48	11	of	of	ADP
cbi-57	48	12	a	a	DET
cbi-57	48	13	given	give	VERB
cbi-57	48	14	length	length	NOUN
cbi-57	48	15	is	be	AUX
cbi-57	48	16	sensitive	sensitive	ADJ
cbi-57	48	17	to	to	ADP
cbi-57	48	18	the	the	DET
cbi-57	48	19	order	order	NOUN
cbi-57	48	20	of	of	ADP
cbi-57	48	21	the	the	DET
cbi-57	48	22	terms	term	NOUN
cbi-57	48	23	in	in	ADP
cbi-57	48	24	the	the	DET
cbi-57	48	25	infinite	infinite	ADJ
cbi-57	48	26	sequence	sequence	NOUN
cbi-57	48	27	.	.	PUNCT
cbi-57	49	1	imagine	imagine	VERB
cbi-57	49	2	summing	sum	VERB
cbi-57	49	3	the	the	DET
cbi-57	49	4	first	first	ADJ
cbi-57	49	5	n	n	PRON
cbi-57	49	6	terms	term	NOUN
cbi-57	49	7	of	of	ADP
cbi-57	49	8	an	an	DET
cbi-57	49	9	infinite	infinite	ADJ
cbi-57	49	10	sequence	sequence	NOUN
cbi-57	49	11	.	.	PUNCT
cbi-57	50	1	the	the	DET
cbi-57	50	2	result	result	NOUN
cbi-57	50	3	will	will	AUX
cbi-57	50	4	not	not	PART
cbi-57	50	5	be	be	AUX
cbi-57	50	6	the	the	DET
cbi-57	50	7	same	same	ADJ
cbi-57	50	8	with	with	ADP
cbi-57	50	9	di!erent	di!erent	NOUN
cbi-57	50	10	orderings	ordering	NOUN
cbi-57	50	11	of	of	ADP
cbi-57	50	12	the	the	DET
cbi-57	50	13	infinite	infinite	ADJ
cbi-57	50	14	sequence	sequence	NOUN
cbi-57	50	15	.	.	PUNCT
cbi-57	51	1	for	for	ADP
cbi-57	51	2	instance	instance	NOUN
cbi-57	51	3	,	,	PUNCT
cbi-57	51	4	if	if	SCONJ
cbi-57	51	5	i	i	PRON
cbi-57	51	6	did	do	VERB
cbi-57	51	7	a	a	DET
cbi-57	51	8	partial	partial	ADJ
cbi-57	51	9	sum	sum	NOUN
cbi-57	51	10	of	of	ADP
cbi-57	51	11	the	the	DET
cbi-57	51	12	first	first	ADJ
cbi-57	51	13	n	n	PRON
cbi-57	51	14	terms	term	NOUN
cbi-57	51	15	of	of	ADP
cbi-57	51	16	an	an	DET
cbi-57	51	17	infinite	infinite	ADJ
cbi-57	51	18	sequence	sequence	NOUN
cbi-57	51	19	,	,	PUNCT
cbi-57	51	20	then	then	ADV
cbi-57	51	21	reversed	reverse	VERB
cbi-57	51	22	the	the	DET
cbi-57	51	23	infinite	infinite	ADJ
cbi-57	51	24	sequence	sequence	NOUN
cbi-57	51	25	,	,	PUNCT
cbi-57	51	26	the	the	DET
cbi-57	51	27	partial	partial	ADJ
cbi-57	51	28	sum	sum	NOUN
cbi-57	51	29	of	of	ADP
cbi-57	51	30	the	the	DET
cbi-57	51	31	first	first	ADJ
cbi-57	51	32	n	n	DET
cbi-57	51	33	terms	term	NOUN
cbi-57	51	34	of	of	ADP
cbi-57	51	35	the	the	DET
cbi-57	51	36	reversed	reverse	VERB
cbi-57	51	37	sequence	sequence	NOUN
cbi-57	51	38	will	will	AUX
cbi-57	51	39	not	not	PART
cbi-57	51	40	necessarily	necessarily	ADV
cbi-57	51	41	match	match	VERB
cbi-57	51	42	the	the	DET
cbi-57	51	43	original	original	ADJ
cbi-57	51	44	partial	partial	ADJ
cbi-57	51	45	sum	sum	NOUN
cbi-57	51	46	.	.	PUNCT
cbi-57	52	1	however	however	ADV
cbi-57	52	2	,	,	PUNCT
cbi-57	52	3	within	within	ADP
cbi-57	52	4	the	the	DET
cbi-57	52	5	first	first	ADJ
cbi-57	52	6	n	n	PRON
cbi-57	52	7	terms	term	NOUN
cbi-57	52	8	,	,	PUNCT
cbi-57	52	9	rearrangements	rearrangement	NOUN
cbi-57	52	10	can	can	AUX
cbi-57	52	11	occur	occur	VERB
cbi-57	52	12	without	without	ADP
cbi-57	52	13	consequence	consequence	NOUN
cbi-57	52	14	.	.	PUNCT
cbi-57	53	1	if	if	SCONJ
cbi-57	53	2	the	the	DET
cbi-57	53	3	extent	extent	NOUN
cbi-57	53	4	of	of	ADP
cbi-57	53	5	the	the	DET
cbi-57	53	6	partial	partial	ADJ
cbi-57	53	7	summation	summation	NOUN
cbi-57	53	8	is	be	AUX
cbi-57	53	9	unknown	unknown	ADJ
cbi-57	53	10	,	,	PUNCT
cbi-57	53	11	then	then	ADV
cbi-57	53	12	it	it	PRON
cbi-57	53	13	is	be	AUX
cbi-57	53	14	also	also	ADV
cbi-57	53	15	unknown	unknown	ADJ
cbi-57	53	16	the	the	DET
cbi-57	53	17	extent	extent	NOUN
cbi-57	53	18	to	to	PART
cbi-57	53	19	which	which	PRON
cbi-57	53	20	numbers	number	NOUN
cbi-57	53	21	can	can	AUX
cbi-57	53	22	be	be	AUX
cbi-57	53	23	reordered	reorder	VERB
cbi-57	53	24	.	.	PUNCT
cbi-57	54	1	for	for	ADP
cbi-57	54	2	the	the	DET
cbi-57	54	3	same	same	ADJ
cbi-57	54	4	reason	reason	NOUN
cbi-57	54	5	,	,	PUNCT
cbi-57	54	6	tacking	tack	VERB
cbi-57	54	7	on	on	ADP
cbi-57	54	8	zeroes	zero	NOUN
cbi-57	54	9	to	to	ADP
cbi-57	54	10	the	the	DET
cbi-57	54	11	beginning	beginning	NOUN
cbi-57	54	12	of	of	ADP
cbi-57	54	13	the	the	DET
cbi-57	54	14	series	series	NOUN
cbi-57	54	15	can	can	AUX
cbi-57	54	16	potentially	potentially	ADV
cbi-57	54	17	change	change	VERB
cbi-57	54	18	the	the	DET
cbi-57	54	19	partial	partial	ADJ
cbi-57	54	20	sum	sum	NOUN
cbi-57	54	21	.	.	PUNCT
cbi-57	55	1	therefore	therefore	ADV
cbi-57	55	2	,	,	PUNCT
cbi-57	55	3	although	although	SCONJ
cbi-57	55	4	adding	add	VERB
cbi-57	55	5	zeroes	zero	NOUN
cbi-57	55	6	to	to	ADP
cbi-57	55	7	the	the	DET
cbi-57	55	8	beginning	beginning	NOUN
cbi-57	55	9	of	of	ADP
cbi-57	55	10	a	a	DET
cbi-57	55	11	series	series	NOUN
cbi-57	55	12	has	have	VERB
cbi-57	55	13	the	the	DET
cbi-57	55	14	appearance	appearance	NOUN
cbi-57	55	15	of	of	ADP
cbi-57	55	16	being	be	AUX
cbi-57	55	17	a	a	DET
cbi-57	55	18	null	null	ADJ
cbi-57	55	19	operation	operation	NOUN
cbi-57	55	20	,	,	PUNCT
cbi-57	55	21	because	because	SCONJ
cbi-57	55	22	doing	do	VERB
cbi-57	55	23	so	so	ADV
cbi-57	55	24	modifies	modify	VERB
cbi-57	55	25	the	the	DET
cbi-57	55	26	value	value	NOUN
cbi-57	55	27	of	of	ADP
cbi-57	55	28	finite	finite	ADJ
cbi-57	55	29	partial	partial	ADJ
cbi-57	55	30	sums	sum	NOUN
cbi-57	55	31	,	,	PUNCT
cbi-57	55	32	it	it	PRON
cbi-57	55	33	can	can	AUX
cbi-57	55	34	also	also	ADV
cbi-57	55	35	lead	lead	VERB
cbi-57	55	36	to	to	ADP
cbi-57	55	37	long	long	ADJ
cbi-57	55	38	-	-	PUNCT
cbi-57	55	39	term	term	NOUN
cbi-57	55	40	changes	change	NOUN
cbi-57	55	41	in	in	ADP
cbi-57	55	42	behavior	behavior	NOUN
cbi-57	55	43	.	.	PUNCT
cbi-57	56	1	additionally	additionally	ADV
cbi-57	56	2	,	,	PUNCT
cbi-57	56	3	changing	change	VERB
cbi-57	56	4	the	the	DET
cbi-57	56	5	number	number	NOUN
cbi-57	56	6	of	of	ADP
cbi-57	56	7	terms	term	NOUN
cbi-57	56	8	in	in	ADP
cbi-57	56	9	a	a	DET
cbi-57	56	10	partial	partial	ADJ
cbi-57	56	11	sum	sum	NOUN
cbi-57	56	12	alters	alter	VERB
cbi-57	56	13	the	the	DET
cbi-57	56	14	value	value	NOUN
cbi-57	56	15	.	.	PUNCT
cbi-57	57	1	adding	add	VERB
cbi-57	57	2	together	together	ADV
cbi-57	57	3	the	the	DET
cbi-57	57	4	first	first	ADJ
cbi-57	57	5	n	n	CCONJ
cbi-57	57	6	numbers	number	NOUN
cbi-57	57	7	of	of	ADP
cbi-57	57	8	a	a	DET
cbi-57	57	9	sequence	sequence	NOUN
cbi-57	57	10	will	will	AUX
cbi-57	57	11	often	often	ADV
cbi-57	57	12	yield	yield	VERB
cbi-57	57	13	a	a	DET
cbi-57	57	14	di!erent	di!erent	NOUN
cbi-57	57	15	value	value	NOUN
cbi-57	57	16	than	than	ADP
cbi-57	57	17	adding	add	VERB
cbi-57	57	18	together	together	ADV
cbi-57	57	19	the	the	DET
cbi-57	57	20	first	first	ADJ
cbi-57	57	21	m	m	NOUN
cbi-57	57	22	numbers	number	NOUN
cbi-57	57	23	of	of	ADP
cbi-57	57	24	the	the	DET
cbi-57	57	25	sequence	sequence	NOUN
cbi-57	57	26	.	.	PUNCT
cbi-57	58	1	in	in	ADP
cbi-57	58	2	short	short	ADJ
cbi-57	58	3	,	,	PUNCT
cbi-57	58	4	partial	partial	ADJ
cbi-57	58	5	sum	sum	NOUN
cbi-57	58	6	behavior	behavior	NOUN
cbi-57	58	7	is	be	AUX
cbi-57	58	8	well	well	ADV
cbi-57	58	9	-	-	PUNCT
cbi-57	58	10	behaved	behave	VERB
cbi-57	58	11	,	,	PUNCT
cbi-57	58	12	wellunderstood	wellunderstood	NOUN
cbi-57	58	13	,	,	PUNCT
cbi-57	58	14	and	and	CCONJ
cbi-57	58	15	well	well	ADV
cbi-57	58	16	-	-	PUNCT
cbi-57	58	17	regulated	regulate	VERB
cbi-57	58	18	.	.	PUNCT
cbi-57	59	1	by	by	ADP
cbi-57	59	2	understanding	understand	VERB
cbi-57	59	3	divergent	divergent	ADJ
cbi-57	59	4	series	series	NOUN
cbi-57	59	5	in	in	ADP
cbi-57	59	6	terms	term	NOUN
cbi-57	59	7	of	of	ADP
cbi-57	59	8	partial	partial	ADJ
cbi-57	59	9	sums	sum	NOUN
cbi-57	59	10	extended	extend	VERB
cbi-57	59	11	into	into	ADP
cbi-57	59	12	the	the	DET
cbi-57	59	13	hyperreals	hyperreal	NOUN
cbi-57	59	14	,	,	PUNCT
cbi-57	59	15	we	we	PRON
cbi-57	59	16	will	will	AUX
cbi-57	59	17	be	be	AUX
cbi-57	59	18	able	able	ADJ
cbi-57	59	19	to	to	PART
cbi-57	59	20	deal	deal	VERB
cbi-57	59	21	with	with	ADP
cbi-57	59	22	them	they	PRON
cbi-57	59	23	more	more	ADV
cbi-57	59	24	rigorously	rigorously	ADV
cbi-57	59	25	and	and	CCONJ
cbi-57	59	26	uniformally	uniformally	ADV
cbi-57	59	27	.	.	PUNCT
cbi-57	60	1	to	to	PART
cbi-57	60	2	understand	understand	VERB
cbi-57	60	3	many	many	ADJ
cbi-57	60	4	of	of	ADP
cbi-57	60	5	the	the	DET
cbi-57	60	6	rules	rule	NOUN
cbi-57	60	7	that	that	PRON
cbi-57	60	8	will	will	AUX
cbi-57	60	9	be	be	AUX
cbi-57	60	10	developed	develop	VERB
cbi-57	60	11	for	for	ADP
cbi-57	60	12	infinite	infinite	ADJ
cbi-57	60	13	series	series	NOUN
cbi-57	60	14	,	,	PUNCT
cbi-57	60	15	imagine	imagine	VERB
cbi-57	60	16	that	that	SCONJ
cbi-57	60	17	the	the	DET
cbi-57	60	18	rules	rule	NOUN
cbi-57	60	19	are	be	AUX
cbi-57	60	20	being	be	AUX
cbi-57	60	21	built	build	VERB
cbi-57	60	22	for	for	ADP
cbi-57	60	23	merely	merely	ADV
cbi-57	60	24	doing	do	VERB
cbi-57	60	25	partial	partial	ADJ
cbi-57	60	26	sums	sum	NOUN
cbi-57	60	27	to	to	ADP
cbi-57	60	28	an	an	DET
cbi-57	60	29	unknown	unknown	ADJ
cbi-57	60	30	parameter	parameter	NOUN
cbi-57	60	31	k	k	NOUN
cbi-57	60	32	,	,	PUNCT
cbi-57	60	33	where	where	SCONJ
cbi-57	60	34	k	k	PROPN
cbi-57	60	35	at	at	ADP
cbi-57	60	36	least	least	ADJ
cbi-57	60	37	acts	act	VERB
cbi-57	60	38	like	like	ADP
cbi-57	60	39	a	a	DET
cbi-57	60	40	particular	particular	ADJ
cbi-57	60	41	finite	finite	NOUN
cbi-57	60	42	value	value	NOUN
cbi-57	60	43	,	,	PUNCT
cbi-57	60	44	but	but	CCONJ
cbi-57	60	45	is	be	AUX
cbi-57	60	46	larger	large	ADJ
cbi-57	60	47	than	than	ADP
cbi-57	60	48	any	any	DET
cbi-57	60	49	particular	particular	ADJ
cbi-57	60	50	list	list	NOUN
cbi-57	60	51	index	index	NOUN
cbi-57	60	52	referenced	reference	VERB
cbi-57	60	53	by	by	ADP
cbi-57	60	54	any	any	DET
cbi-57	60	55	finite	finite	ADJ
cbi-57	60	56	manipulation	manipulation	NOUN
cbi-57	60	57	of	of	ADP
cbi-57	60	58	the	the	DET
cbi-57	60	59	series	series	NOUN
cbi-57	60	60	.	.	PUNCT
cbi-57	61	1	some	some	PRON
cbi-57	61	2	of	of	ADP
cbi-57	61	3	these	these	DET
cbi-57	61	4	formulas	formula	NOUN
cbi-57	61	5	will	will	AUX
cbi-57	61	6	be	be	AUX
cbi-57	61	7	further	far	ADV
cbi-57	61	8	reducible	reducible	ADJ
cbi-57	61	9	due	due	ADP
cbi-57	61	10	to	to	ADP
cbi-57	61	11	the	the	DET
cbi-57	61	12	nature	nature	NOUN
cbi-57	61	13	of	of	ADP
cbi-57	61	14	the	the	DET
cbi-57	61	15	hyperreals	hyperreal	NOUN
cbi-57	61	16	,	,	PUNCT
cbi-57	61	17	as	as	SCONJ
cbi-57	61	18	will	will	AUX
cbi-57	61	19	be	be	AUX
cbi-57	61	20	discussed	discuss	VERB
cbi-57	61	21	in	in	ADP
cbi-57	61	22	section	section	NOUN
cbi-57	61	23	7	7	NUM
cbi-57	61	24	.	.	NOUN
cbi-57	61	25	4	4	NUM
cbi-57	61	26	pinning	pin	VERB
cbi-57	61	27	down	down	ADP
cbi-57	61	28	!	!	PUNCT
cbi-57	62	1	since	since	SCONJ
cbi-57	62	2	!	!	PUNCT
cbi-57	62	3	operates	operate	VERB
cbi-57	62	4	as	as	ADP
cbi-57	62	5	a	a	DET
cbi-57	62	6	benchmark	benchmark	NOUN
cbi-57	62	7	instead	instead	ADV
cbi-57	62	8	of	of	ADP
cbi-57	62	9	a	a	DET
cbi-57	62	10	number	number	NOUN
cbi-57	62	11	,	,	PUNCT
cbi-57	62	12	the	the	DET
cbi-57	62	13	first	first	ADJ
cbi-57	62	14	task	task	NOUN
cbi-57	62	15	is	be	AUX
cbi-57	62	16	to	to	PART
cbi-57	62	17	identify	identify	VERB
cbi-57	62	18	the	the	DET
cbi-57	62	19	benchmark	benchmark	NOUN
cbi-57	62	20	to	to	PART
cbi-57	62	21	associate	associate	VERB
cbi-57	62	22	!	!	PUNCT
cbi-57	63	1	with	with	ADP
cbi-57	63	2	.	.	PUNCT
cbi-57	64	1	this	this	PRON
cbi-57	64	2	is	be	AUX
cbi-57	64	3	actually	actually	ADV
cbi-57	64	4	to	to	ADP
cbi-57	64	5	some	some	DET
cbi-57	64	6	extent	extent	NOUN
cbi-57	64	7	an	an	DET
cbi-57	64	8	arbitrary	arbitrary	ADJ
cbi-57	64	9	decision	decision	NOUN
cbi-57	64	10	.	.	PUNCT
cbi-57	65	1	any	any	DET
cbi-57	65	2	infinitely	infinitely	ADV
cbi-57	65	3	large	large	ADJ
cbi-57	65	4	value	value	NOUN
cbi-57	65	5	could	could	AUX
cbi-57	65	6	be	be	AUX
cbi-57	65	7	used	use	VERB
cbi-57	65	8	to	to	PART
cbi-57	65	9	establish	establish	VERB
cbi-57	65	10	a	a	DET
cbi-57	65	11	baseline	baseline	NOUN
cbi-57	65	12	!	!	PUNCT
cbi-57	65	13	.	.	PUNCT
cbi-57	66	1	however	however	ADV
cbi-57	66	2	,	,	PUNCT
cbi-57	66	3	the	the	DET
cbi-57	66	4	value	value	NOUN
cbi-57	66	5	that	that	PRON
cbi-57	66	6	seems	seem	VERB
cbi-57	66	7	most	most	ADV
cbi-57	66	8	natural	natural	ADJ
cbi-57	66	9	for	for	ADP
cbi-57	66	10	!	!	PUNCT
cbi-57	67	1	(	(	PUNCT
cbi-57	67	2	especially	especially	ADV
cbi-57	67	3	for	for	ADP
cbi-57	67	4	summation	summation	NOUN
cbi-57	67	5	)	)	PUNCT
cbi-57	67	6	is	be	AUX
cbi-57	67	7	the	the	DET
cbi-57	67	8	size	size	NOUN
cbi-57	67	9	of	of	ADP
cbi-57	67	10	the	the	DET
cbi-57	67	11	set	set	NOUN
cbi-57	67	12	of	of	ADP
cbi-57	67	13	positive	positive	ADJ
cbi-57	67	14	integers	integer	NOUN
cbi-57	67	15	.	.	PUNCT
cbi-57	68	1	therefore	therefore	ADV
cbi-57	68	2	,	,	PUNCT
cbi-57	68	3	!	!	PUNCT
cbi-57	68	4	will	will	AUX
cbi-57	68	5	be	be	AUX
cbi-57	68	6	used	use	VERB
cbi-57	68	7	to	to	PART
cbi-57	68	8	refer	refer	VERB
cbi-57	68	9	to	to	ADP
cbi-57	68	10	the	the	DET
cbi-57	68	11	total	total	ADJ
cbi-57	68	12	quantity	quantity	NOUN
cbi-57	68	13	of	of	ADP
cbi-57	68	14	positive	positive	ADJ
cbi-57	68	15	integers.3	integers.3	NOUN
cbi-57	68	16	!	!	PUNCT
cbi-57	69	1	=	=	PUNCT
cbi-57	70	1	|n|	|n|	NUM
cbi-57	70	2	=	=	PUNCT
cbi-57	70	3	!	!	PUNCT
cbi-57	71	1	i!n	i!n	PROPN
cbi-57	71	2	1	1	NUM
cbi-57	71	3	(	(	PUNCT
cbi-57	71	4	1	1	NUM
cbi-57	71	5	)	)	PUNCT
cbi-57	71	6	because	because	SCONJ
cbi-57	71	7	of	of	ADP
cbi-57	71	8	this	this	PRON
cbi-57	71	9	,	,	PUNCT
cbi-57	71	10	the	the	DET
cbi-57	71	11	notation	notation	NOUN
cbi-57	71	12	used	use	VERB
cbi-57	71	13	will	will	AUX
cbi-57	71	14	be	be	AUX
cbi-57	71	15	more	more	ADV
cbi-57	71	16	specific	specific	ADJ
cbi-57	71	17	when	when	SCONJ
cbi-57	71	18	writing	write	VERB
cbi-57	71	19	summations	summation	NOUN
cbi-57	71	20	.	.	PUNCT
cbi-57	72	1	instead	instead	ADV
cbi-57	72	2	of	of	ADP
cbi-57	72	3	summing	sum	VERB
cbi-57	72	4	to	to	ADP
cbi-57	72	5	the	the	DET
cbi-57	72	6	ambiguous	ambiguous	ADJ
cbi-57	72	7	infinity	infinity	NOUN
cbi-57	72	8	,	,	PUNCT
cbi-57	72	9	"	"	PUNCT
cbi-57	72	10	,	,	PUNCT
cbi-57	72	11	a	a	DET
cbi-57	72	12	summation	summation	NOUN
cbi-57	72	13	to	to	ADP
cbi-57	72	14	the	the	DET
cbi-57	72	15	specific	specific	ADJ
cbi-57	72	16	infinity	infinity	NOUN
cbi-57	72	17	of	of	ADP
cbi-57	72	18	all	all	DET
cbi-57	72	19	positive	positive	ADJ
cbi-57	72	20	integers	integer	NOUN
cbi-57	72	21	,	,	PUNCT
cbi-57	72	22	!	!	PUNCT
cbi-57	72	23	,	,	PUNCT
cbi-57	72	24	will	will	AUX
cbi-57	72	25	be	be	AUX
cbi-57	72	26	used	use	VERB
cbi-57	72	27	.	.	PUNCT
cbi-57	73	1	therefore	therefore	ADV
cbi-57	73	2	,	,	PUNCT
cbi-57	73	3	the	the	DET
cbi-57	73	4	series	series	NOUN
cbi-57	73	5	1	1	NUM
cbi-57	73	6	+	+	SYM
cbi-57	73	7	2	2	NUM
cbi-57	73	8	+	+	NUM
cbi-57	73	9	3	3	NUM
cbi-57	73	10	+	+	NUM
cbi-57	73	11	.	.	PUNCT
cbi-57	73	12	.	.	PUNCT
cbi-57	73	13	.	.	PUNCT
cbi-57	74	1	will	will	AUX
cbi-57	74	2	be	be	AUX
cbi-57	74	3	written	write	VERB
cbi-57	74	4	as	as	ADP
cbi-57	74	5	!	!	PUNCT
cbi-57	74	6	!	!	PUNCT
cbi-57	75	1	i=1	i=1	PROPN
cbi-57	76	1	i	i	PRON
cbi-57	76	2	(	(	PUNCT
cbi-57	76	3	2	2	NUM
cbi-57	76	4	)	)	PUNCT
cbi-57	76	5	(	(	PUNCT
cbi-57	76	6	1	1	X
cbi-57	76	7	)	)	PUNCT
cbi-57	76	8	will	will	AUX
cbi-57	76	9	establish	establish	VERB
cbi-57	76	10	the	the	DET
cbi-57	76	11	starting	starting	NOUN
cbi-57	76	12	benchmark	benchmark	NOUN
cbi-57	76	13	for	for	ADP
cbi-57	76	14	relationships	relationship	NOUN
cbi-57	76	15	among	among	ADP
cbi-57	76	16	the	the	DET
cbi-57	76	17	di!erent	di!erent	NOUN
cbi-57	76	18	series	series	NOUN
cbi-57	76	19	.	.	PUNCT
cbi-57	77	1	5	5	NUM
cbi-57	77	2	the	the	DET
cbi-57	77	3	standard	standard	ADJ
cbi-57	77	4	summation	summation	NOUN
cbi-57	77	5	because	because	SCONJ
cbi-57	77	6	partial	partial	ADJ
cbi-57	77	7	sums	sum	NOUN
cbi-57	77	8	are	be	AUX
cbi-57	77	9	dependent	dependent	ADJ
cbi-57	77	10	on	on	ADP
cbi-57	77	11	length	length	NOUN
cbi-57	77	12	and	and	CCONJ
cbi-57	77	13	order	order	NOUN
cbi-57	77	14	,	,	PUNCT
cbi-57	77	15	it	it	PRON
cbi-57	77	16	is	be	AUX
cbi-57	77	17	important	important	ADJ
cbi-57	77	18	to	to	PART
cbi-57	77	19	establish	establish	VERB
cbi-57	77	20	an	an	DET
cbi-57	77	21	o"cial	o"cial	ADJ
cbi-57	77	22	standardization	standardization	NOUN
cbi-57	77	23	of	of	ADP
cbi-57	77	24	summation	summation	NOUN
cbi-57	77	25	.	.	PUNCT
cbi-57	78	1	that	that	PRON
cbi-57	78	2	is	be	AUX
cbi-57	78	3	,	,	PUNCT
cbi-57	78	4	!	!	PUNCT
cbi-57	78	5	"	"	PUNCT
cbi-57	79	1	i=1	i=1	PROPN
cbi-57	79	2	will	will	AUX
cbi-57	79	3	be	be	AUX
cbi-57	79	4	di!erent	di!erent	VERB
cbi-57	79	5	from	from	ADP
cbi-57	79	6	!	!	PUNCT
cbi-57	79	7	"	"	PUNCT
cbi-57	80	1	i=0	i=0	PROPN
cbi-57	80	2	.	.	PUNCT
cbi-57	81	1	even	even	ADV
cbi-57	81	2	though	though	SCONJ
cbi-57	81	3	it	it	PRON
cbi-57	81	4	looks	look	VERB
cbi-57	81	5	like	like	ADP
cbi-57	81	6	series	series	NOUN
cbi-57	81	7	with	with	ADP
cbi-57	81	8	these	these	DET
cbi-57	81	9	types	type	NOUN
cbi-57	81	10	of	of	ADP
cbi-57	81	11	sums	sum	NOUN
cbi-57	81	12	will	will	AUX
cbi-57	81	13	have	have	VERB
cbi-57	81	14	an	an	DET
cbi-57	81	15	identical	identical	ADJ
cbi-57	81	16	number	number	NOUN
cbi-57	81	17	of	of	ADP
cbi-57	81	18	terms	term	NOUN
cbi-57	81	19	(	(	PUNCT
cbi-57	81	20	after	after	ADV
cbi-57	81	21	all	all	PRON
cbi-57	81	22	they	they	PRON
cbi-57	81	23	both	both	PRON
cbi-57	81	24	have	have	VERB
cbi-57	81	25	infinite	infinite	ADJ
cbi-57	81	26	terms	term	NOUN
cbi-57	81	27	)	)	PUNCT
cbi-57	81	28	,	,	PUNCT
cbi-57	81	29	using	use	VERB
cbi-57	81	30	this	this	DET
cbi-57	81	31	methodology	methodology	NOUN
cbi-57	81	32	the	the	DET
cbi-57	81	33	latter	latter	ADJ
cbi-57	81	34	one	one	NOUN
cbi-57	81	35	will	will	AUX
cbi-57	81	36	actually	actually	ADV
cbi-57	81	37	have	have	VERB
cbi-57	81	38	more	more	ADJ
cbi-57	81	39	elements	element	NOUN
cbi-57	81	40	than	than	ADP
cbi-57	81	41	the	the	DET
cbi-57	81	42	former	former	NOUN
cbi-57	81	43	.	.	PUNCT
cbi-57	82	1	this	this	PRON
cbi-57	82	2	is	be	AUX
cbi-57	82	3	due	due	ADJ
cbi-57	82	4	to	to	ADP
cbi-57	82	5	the	the	DET
cbi-57	82	6	principle	principle	NOUN
cbi-57	82	7	established	establish	VERB
cbi-57	82	8	in	in	ADP
cbi-57	82	9	section	section	NOUN
cbi-57	82	10	3	3	NUM
cbi-57	82	11	.	.	PUNCT
cbi-57	83	1	if	if	SCONJ
cbi-57	83	2	,	,	PUNCT
cbi-57	83	3	instead	instead	ADV
cbi-57	83	4	of	of	ADP
cbi-57	83	5	!	!	PUNCT
cbi-57	84	1	being	be	AUX
cbi-57	84	2	infinite	infinite	ADJ
cbi-57	84	3	,	,	PUNCT
cbi-57	84	4	pretend	pretend	VERB
cbi-57	84	5	that	that	PRON
cbi-57	84	6	!	!	PUNCT
cbi-57	85	1	was	be	AUX
cbi-57	85	2	just	just	ADV
cbi-57	85	3	an	an	DET
cbi-57	85	4	ordinary	ordinary	ADJ
cbi-57	85	5	finite	finite	NOUN
cbi-57	85	6	integer	integer	NOUN
cbi-57	85	7	parameter	parameter	NOUN
cbi-57	85	8	.	.	PUNCT
cbi-57	86	1	examine	examine	VERB
cbi-57	86	2	the	the	DET
cbi-57	86	3	series	series	NOUN
cbi-57	86	4	!	!	PUNCT
cbi-57	86	5	!	!	PUNCT
cbi-57	87	1	i=1	i=1	PROPN
cbi-57	88	1	1	1	X
cbi-57	88	2	.	.	PUNCT
cbi-57	88	3	(	(	PUNCT
cbi-57	88	4	3	3	X
cbi-57	88	5	)	)	PUNCT
cbi-57	88	6	3there	3there	NUM
cbi-57	88	7	are	be	AUX
cbi-57	88	8	some	some	DET
cbi-57	88	9	objections	objection	NOUN
cbi-57	88	10	to	to	ADP
cbi-57	88	11	equating	equate	VERB
cbi-57	88	12	a	a	DET
cbi-57	88	13	hyperreal	hyperreal	ADJ
cbi-57	88	14	number	number	NOUN
cbi-57	88	15	to	to	ADP
cbi-57	88	16	a	a	DET
cbi-57	88	17	cardinal	cardinal	ADJ
cbi-57	88	18	number	number	NOUN
cbi-57	88	19	such	such	ADJ
cbi-57	88	20	as	as	ADP
cbi-57	88	21	|n	|n	NOUN
cbi-57	88	22	|	|	NOUN
cbi-57	88	23	.	.	PUNCT
cbi-57	89	1	the	the	DET
cbi-57	89	2	specific	specific	ADJ
cbi-57	89	3	identity	identity	NOUN
cbi-57	89	4	of	of	ADP
cbi-57	89	5	!	!	PUNCT
cbi-57	90	1	with	with	ADP
cbi-57	90	2	|n	|n	NOUN
cbi-57	90	3	|	|	ADV
cbi-57	90	4	is	be	AUX
cbi-57	90	5	for	for	ADP
cbi-57	90	6	conceptual	conceptual	ADJ
cbi-57	90	7	convenience	convenience	NOUN
cbi-57	90	8	.	.	PUNCT
cbi-57	91	1	alternatively	alternatively	ADV
cbi-57	91	2	,	,	PUNCT
cbi-57	91	3	simply	simply	ADV
cbi-57	91	4	treating	treat	VERB
cbi-57	91	5	!	!	PUNCT
cbi-57	92	1	as	as	ADP
cbi-57	92	2	an	an	DET
cbi-57	92	3	arbitrary	arbitrary	ADJ
cbi-57	92	4	(	(	PUNCT
cbi-57	92	5	but	but	CCONJ
cbi-57	92	6	unchanging	unchanging	ADJ
cbi-57	92	7	)	)	PUNCT
cbi-57	92	8	benchmark	benchmark	NOUN
cbi-57	92	9	of	of	ADP
cbi-57	92	10	infinity	infinity	NOUN
cbi-57	92	11	leads	lead	VERB
cbi-57	92	12	to	to	ADP
cbi-57	92	13	the	the	DET
cbi-57	92	14	same	same	ADJ
cbi-57	92	15	results	result	NOUN
cbi-57	92	16	.	.	PUNCT
cbi-57	93	1	volume	volume	NOUN
cbi-57	93	2	2	2	NUM
cbi-57	93	3	,	,	PUNCT
cbi-57	93	4	issue	issue	NOUN
cbi-57	93	5	1	1	NUM
cbi-57	93	6	6	6	NUM
cbi-57	93	7	.	.	PUNCT
cbi-57	93	8	simple	simple	ADJ
cbi-57	93	9	arithmetic	arithmetic	ADJ
cbi-57	93	10	and	and	CCONJ
cbi-57	93	11	geometric	geometric	ADJ
cbi-57	93	12	series	series	NOUN
cbi-57	93	13	9	9	NUM
cbi-57	93	14	if	if	SCONJ
cbi-57	93	15	!	!	PUNCT
cbi-57	93	16	represented	represent	VERB
cbi-57	93	17	an	an	DET
cbi-57	93	18	integer	integer	NOUN
cbi-57	93	19	(	(	PUNCT
cbi-57	93	20	say	say	INTJ
cbi-57	93	21	,	,	PUNCT
cbi-57	93	22	5	5	NUM
cbi-57	93	23	)	)	PUNCT
cbi-57	93	24	instead	instead	ADV
cbi-57	93	25	of	of	ADP
cbi-57	93	26	"	"	PUNCT
cbi-57	93	27	,	,	PUNCT
cbi-57	93	28	it	it	PRON
cbi-57	93	29	would	would	AUX
cbi-57	93	30	be	be	AUX
cbi-57	93	31	obvious	obvious	ADJ
cbi-57	93	32	that	that	SCONJ
cbi-57	93	33	this	this	DET
cbi-57	93	34	sum	sum	NOUN
cbi-57	93	35	represents	represent	VERB
cbi-57	93	36	a	a	DET
cbi-57	93	37	di!erent	di!erent	NOUN
cbi-57	93	38	value	value	NOUN
cbi-57	93	39	from	from	ADP
cbi-57	93	40	the	the	DET
cbi-57	93	41	series	series	NOUN
cbi-57	93	42	!	!	PUNCT
cbi-57	93	43	!	!	PUNCT
cbi-57	94	1	i=0	i=0	PROPN
cbi-57	94	2	1	1	NUM
cbi-57	94	3	.	.	PUNCT
cbi-57	94	4	(	(	PUNCT
cbi-57	94	5	4	4	X
cbi-57	94	6	)	)	PUNCT
cbi-57	94	7	equation	equation	NOUN
cbi-57	94	8	3	3	NUM
cbi-57	94	9	would	would	AUX
cbi-57	94	10	represent	represent	VERB
cbi-57	94	11	the	the	DET
cbi-57	94	12	value	value	NOUN
cbi-57	94	13	5	5	NUM
cbi-57	94	14	while	while	SCONJ
cbi-57	94	15	equation	equation	NOUN
cbi-57	94	16	4	4	NUM
cbi-57	94	17	would	would	AUX
cbi-57	94	18	represent	represent	VERB
cbi-57	94	19	the	the	DET
cbi-57	94	20	value	value	NOUN
cbi-57	94	21	6	6	NUM
cbi-57	94	22	.	.	PUNCT
cbi-57	95	1	therefore	therefore	ADV
cbi-57	95	2	,	,	PUNCT
cbi-57	95	3	it	it	PRON
cbi-57	95	4	is	be	AUX
cbi-57	95	5	clear	clear	ADJ
cbi-57	95	6	that	that	SCONJ
cbi-57	95	7	having	have	VERB
cbi-57	95	8	matching	match	VERB
cbi-57	95	9	indices	index	NOUN
cbi-57	95	10	matters	matter	NOUN
cbi-57	95	11	.	.	PUNCT
cbi-57	96	1	in	in	ADP
cbi-57	96	2	fact	fact	NOUN
cbi-57	96	3	,	,	PUNCT
cbi-57	96	4	our	our	PRON
cbi-57	96	5	ability	ability	NOUN
cbi-57	96	6	to	to	PART
cbi-57	96	7	sum	sum	VERB
cbi-57	96	8	divergent	divergent	ADJ
cbi-57	96	9	series	series	NOUN
cbi-57	96	10	will	will	AUX
cbi-57	96	11	sometimes	sometimes	ADV
cbi-57	96	12	depend	depend	VERB
cbi-57	96	13	on	on	ADP
cbi-57	96	14	having	have	VERB
cbi-57	96	15	summations	summation	NOUN
cbi-57	96	16	with	with	ADP
cbi-57	96	17	equivalent	equivalent	ADJ
cbi-57	96	18	numbers	number	NOUN
cbi-57	96	19	of	of	ADP
cbi-57	96	20	terms	term	NOUN
cbi-57	96	21	.	.	PUNCT
cbi-57	97	1	therefore	therefore	ADV
cbi-57	97	2	,	,	PUNCT
cbi-57	97	3	a	a	DET
cbi-57	97	4	“	"	PUNCT
cbi-57	97	5	standard	standard	NOUN
cbi-57	97	6	”	"	PUNCT
cbi-57	97	7	starting	starting	NOUN
cbi-57	97	8	point	point	NOUN
cbi-57	97	9	for	for	ADP
cbi-57	97	10	summation	summation	NOUN
cbi-57	97	11	will	will	AUX
cbi-57	97	12	need	need	VERB
cbi-57	97	13	to	to	PART
cbi-57	97	14	be	be	AUX
cbi-57	97	15	established	establish	VERB
cbi-57	97	16	in	in	ADP
cbi-57	97	17	order	order	NOUN
cbi-57	97	18	to	to	PART
cbi-57	97	19	ensure	ensure	VERB
cbi-57	97	20	that	that	SCONJ
cbi-57	97	21	like	like	ADP
cbi-57	97	22	entities	entity	NOUN
cbi-57	97	23	are	be	AUX
cbi-57	97	24	being	be	AUX
cbi-57	97	25	compared	compare	VERB
cbi-57	97	26	and	and	CCONJ
cbi-57	97	27	reasoned	reason	VERB
cbi-57	97	28	about	about	ADP
cbi-57	97	29	.	.	PUNCT
cbi-57	98	1	since	since	SCONJ
cbi-57	98	2	!	!	PUNCT
cbi-57	98	3	has	have	AUX
cbi-57	98	4	been	be	AUX
cbi-57	98	5	defined	define	VERB
cbi-57	98	6	as	as	ADP
cbi-57	98	7	being	be	AUX
cbi-57	98	8	the	the	DET
cbi-57	98	9	size	size	NOUN
cbi-57	98	10	of	of	ADP
cbi-57	98	11	the	the	DET
cbi-57	98	12	set	set	NOUN
cbi-57	98	13	of	of	ADP
cbi-57	98	14	all	all	DET
cbi-57	98	15	positive	positive	ADJ
cbi-57	98	16	integers	integer	NOUN
cbi-57	98	17	,	,	PUNCT
cbi-57	98	18	it	it	PRON
cbi-57	98	19	makes	make	VERB
cbi-57	98	20	sense	sense	NOUN
cbi-57	98	21	to	to	PART
cbi-57	98	22	start	start	VERB
cbi-57	98	23	at	at	ADP
cbi-57	98	24	1	1	NUM
cbi-57	98	25	.	.	PUNCT
cbi-57	99	1	for	for	ADP
cbi-57	99	2	the	the	DET
cbi-57	99	3	purposes	purpose	NOUN
cbi-57	99	4	of	of	ADP
cbi-57	99	5	this	this	DET
cbi-57	99	6	paper	paper	NOUN
cbi-57	99	7	,	,	PUNCT
cbi-57	99	8	the	the	DET
cbi-57	99	9	“	"	PUNCT
cbi-57	99	10	standard	standard	ADJ
cbi-57	99	11	”	"	PUNCT
cbi-57	99	12	way	way	NOUN
cbi-57	99	13	of	of	ADP
cbi-57	99	14	summing	summing	NOUN
cbi-57	99	15	will	will	AUX
cbi-57	99	16	be	be	AUX
cbi-57	99	17	to	to	PART
cbi-57	99	18	start	start	VERB
cbi-57	99	19	with	with	ADP
cbi-57	99	20	1	1	NUM
cbi-57	99	21	and	and	CCONJ
cbi-57	99	22	proceed	proceed	VERB
cbi-57	99	23	to	to	ADP
cbi-57	99	24	!	!	PUNCT
cbi-57	99	25	.	.	PUNCT
cbi-57	100	1	6	6	NUM
cbi-57	100	2	simple	simple	ADJ
cbi-57	100	3	arithmetic	arithmetic	ADJ
cbi-57	100	4	and	and	CCONJ
cbi-57	100	5	geometric	geometric	ADJ
cbi-57	100	6	series	series	NOUN
cbi-57	100	7	6.1	6.1	NUM
cbi-57	100	8	arithmetic	arithmetic	PROPN
cbi-57	100	9	series	series	PROPN
cbi-57	100	10	arithmetic	arithmetic	PROPN
cbi-57	100	11	series	series	PROPN
cbi-57	100	12	take	take	VERB
cbi-57	100	13	the	the	DET
cbi-57	100	14	form	form	NOUN
cbi-57	100	15	n	n	CCONJ
cbi-57	100	16	!	!	PUNCT
cbi-57	101	1	i=1	i=1	PROPN
cbi-57	102	1	a	a	PRON
cbi-57	103	1	+	+	NOUN
cbi-57	103	2	(	(	PUNCT
cbi-57	103	3	i	i	NOUN
cbi-57	103	4	#	#	NOUN
cbi-57	103	5	1)d	1)d	NUM
cbi-57	103	6	.	.	PUNCT
cbi-57	104	1	(	(	PUNCT
cbi-57	104	2	5	5	X
cbi-57	104	3	)	)	PUNCT
cbi-57	104	4	the	the	DET
cbi-57	104	5	sum	sum	NOUN
cbi-57	104	6	of	of	ADP
cbi-57	104	7	an	an	DET
cbi-57	104	8	arithmetic	arithmetic	ADJ
cbi-57	104	9	series	series	NOUN
cbi-57	104	10	,	,	PUNCT
cbi-57	104	11	given	give	VERB
cbi-57	104	12	a	a	DET
cbi-57	104	13	starting	starting	NOUN
cbi-57	104	14	value	value	NOUN
cbi-57	104	15	a	a	PRON
cbi-57	104	16	,	,	PUNCT
cbi-57	104	17	the	the	DET
cbi-57	104	18	number	number	NOUN
cbi-57	104	19	of	of	ADP
cbi-57	104	20	elements	element	NOUN
cbi-57	104	21	n	n	CCONJ
cbi-57	104	22	,	,	PUNCT
cbi-57	104	23	and	and	CCONJ
cbi-57	104	24	distance	distance	NOUN
cbi-57	104	25	between	between	ADP
cbi-57	104	26	elements	element	NOUN
cbi-57	104	27	d	d	NOUN
cbi-57	104	28	,	,	PUNCT
cbi-57	104	29	can	can	AUX
cbi-57	104	30	be	be	AUX
cbi-57	104	31	given	give	VERB
cbi-57	104	32	by	by	ADP
cbi-57	104	33	the	the	DET
cbi-57	104	34	formula	formula	NOUN
cbi-57	104	35	n	n	NOUN
cbi-57	104	36	!	!	PUNCT
cbi-57	105	1	i=1	i=1	PROPN
cbi-57	106	1	a	a	DET
cbi-57	106	2	+	+	NOUN
cbi-57	106	3	(	(	PUNCT
cbi-57	106	4	i	i	NOUN
cbi-57	106	5	#	#	NOUN
cbi-57	106	6	1)d	1)d	NUM
cbi-57	106	7	=	=	SYM
cbi-57	106	8	n	n	PRON
cbi-57	106	9	2	2	NUM
cbi-57	106	10	(	(	PUNCT
cbi-57	106	11	2a	2a	NUM
cbi-57	106	12	+	+	CCONJ
cbi-57	106	13	(	(	PUNCT
cbi-57	106	14	n	n	PRON
cbi-57	106	15	#	#	NOUN
cbi-57	106	16	1)d	1)d	NUM
cbi-57	106	17	)	)	PUNCT
cbi-57	106	18	.	.	PUNCT
cbi-57	107	1	(	(	PUNCT
cbi-57	107	2	6	6	NUM
cbi-57	107	3	)	)	PUNCT
cbi-57	107	4	to	to	PART
cbi-57	107	5	find	find	VERB
cbi-57	107	6	the	the	DET
cbi-57	107	7	sum	sum	NOUN
cbi-57	107	8	of	of	ADP
cbi-57	107	9	an	an	DET
cbi-57	107	10	infinite	infinite	ADJ
cbi-57	107	11	arithmetic	arithmetic	ADJ
cbi-57	107	12	series	series	NOUN
cbi-57	107	13	,	,	PUNCT
cbi-57	107	14	!	!	PUNCT
cbi-57	108	1	is	be	AUX
cbi-57	108	2	used	use	VERB
cbi-57	108	3	for	for	ADP
cbi-57	108	4	n	n	NUM
cbi-57	108	5	,	,	PUNCT
cbi-57	108	6	forming	form	VERB
cbi-57	108	7	a	a	DET
cbi-57	108	8	hyperreal	hyperreal	NOUN
cbi-57	108	9	value	value	NOUN
cbi-57	108	10	.	.	PUNCT
cbi-57	109	1	that	that	PRON
cbi-57	109	2	reduces	reduce	VERB
cbi-57	109	3	the	the	DET
cbi-57	109	4	formula	formula	NOUN
cbi-57	109	5	to	to	ADP
cbi-57	109	6	!	!	PUNCT
cbi-57	109	7	!	!	PUNCT
cbi-57	110	1	i=1	i=1	PROPN
cbi-57	111	1	a	a	PRON
cbi-57	112	1	+	+	NOUN
cbi-57	112	2	(	(	PUNCT
cbi-57	112	3	i	i	NOUN
cbi-57	112	4	#	#	NOUN
cbi-57	112	5	1)d	1)d	NUM
cbi-57	112	6	=	=	PUNCT
cbi-57	112	7	!	!	PUNCT
cbi-57	113	1	a	a	PRON
cbi-57	113	2	+	+	NOUN
cbi-57	113	3	!	!	PUNCT
cbi-57	113	4	2d	2d	NOUN
cbi-57	113	5	2	2	NUM
cbi-57	113	6	#	#	NOUN
cbi-57	113	7	!	!	PUNCT
cbi-57	114	1	d	d	X
cbi-57	114	2	2	2	NUM
cbi-57	114	3	.	.	PUNCT
cbi-57	115	1	(	(	PUNCT
cbi-57	115	2	7	7	NUM
cbi-57	115	3	)	)	PUNCT
cbi-57	115	4	therefore	therefore	ADV
cbi-57	115	5	,	,	PUNCT
cbi-57	115	6	to	to	PART
cbi-57	115	7	find	find	VERB
cbi-57	115	8	the	the	DET
cbi-57	115	9	summation	summation	NOUN
cbi-57	115	10	of	of	ADP
cbi-57	115	11	the	the	DET
cbi-57	115	12	series	series	NOUN
cbi-57	115	13	1	1	NUM
cbi-57	115	14	+	+	PROPN
cbi-57	115	15	1	1	NUM
cbi-57	115	16	+	+	SYM
cbi-57	115	17	1	1	NUM
cbi-57	115	18	+	+	NUM
cbi-57	115	19	.	.	PUNCT
cbi-57	115	20	.	.	PUNCT
cbi-57	116	1	.	.	PUNCT
cbi-57	116	2	,	,	PUNCT
cbi-57	116	3	one	one	PRON
cbi-57	116	4	must	must	AUX
cbi-57	116	5	only	only	ADV
cbi-57	116	6	substitute	substitute	VERB
cbi-57	116	7	in	in	ADP
cbi-57	116	8	the	the	DET
cbi-57	116	9	correct	correct	ADJ
cbi-57	116	10	parameters	parameter	NOUN
cbi-57	116	11	.	.	PUNCT
cbi-57	117	1	since	since	SCONJ
cbi-57	117	2	the	the	DET
cbi-57	117	3	starting	starting	NOUN
cbi-57	117	4	value	value	NOUN
cbi-57	117	5	is	be	AUX
cbi-57	117	6	1	1	NUM
cbi-57	117	7	and	and	CCONJ
cbi-57	117	8	the	the	DET
cbi-57	117	9	distance	distance	NOUN
cbi-57	117	10	between	between	ADP
cbi-57	117	11	terms	term	NOUN
cbi-57	117	12	is	be	AUX
cbi-57	117	13	0	0	NUM
cbi-57	117	14	,	,	PUNCT
cbi-57	117	15	this	this	DET
cbi-57	117	16	yields	yield	NOUN
cbi-57	117	17	!	!	PUNCT
cbi-57	117	18	!	!	PUNCT
cbi-57	118	1	i=1	i=1	PROPN
cbi-57	119	1	1	1	NUM
cbi-57	119	2	=	=	PUNCT
cbi-57	119	3	!	!	PUNCT
cbi-57	119	4	·	·	PUNCT
cbi-57	120	1	1	1	NUM
cbi-57	121	1	+	+	CCONJ
cbi-57	121	2	!	!	NUM
cbi-57	122	1	2	2	X
cbi-57	122	2	·	·	SYM
cbi-57	122	3	0	0	NUM
cbi-57	122	4	2	2	NUM
cbi-57	122	5	#	#	NOUN
cbi-57	122	6	!	!	PUNCT
cbi-57	122	7	·	·	PUNCT
cbi-57	123	1	0	0	NUM
cbi-57	123	2	2	2	NUM
cbi-57	123	3	(	(	PUNCT
cbi-57	123	4	8)	8)	NUM
cbi-57	123	5	=	=	PUNCT
cbi-57	123	6	!	!	PUNCT
cbi-57	124	1	+	+	CCONJ
cbi-57	124	2	0	0	NUM
cbi-57	124	3	#	#	NOUN
cbi-57	124	4	0	0	NUM
cbi-57	125	1	(	(	PUNCT
cbi-57	125	2	9	9	NUM
cbi-57	125	3	)	)	PUNCT
cbi-57	125	4	=	=	PUNCT
cbi-57	125	5	!	!	PUNCT
cbi-57	125	6	.	.	PUNCT
cbi-57	126	1	(	(	PUNCT
cbi-57	126	2	10	10	NUM
cbi-57	126	3	)	)	PUNCT
cbi-57	126	4	it	it	PRON
cbi-57	126	5	is	be	AUX
cbi-57	126	6	intuitively	intuitively	ADV
cbi-57	126	7	obvious	obvious	ADJ
cbi-57	126	8	that	that	SCONJ
cbi-57	126	9	since	since	SCONJ
cbi-57	126	10	there	there	PRON
cbi-57	126	11	are	be	VERB
cbi-57	126	12	!	!	PUNCT
cbi-57	127	1	1s	1	NOUN
cbi-57	127	2	added	add	VERB
cbi-57	127	3	together	together	ADV
cbi-57	127	4	that	that	SCONJ
cbi-57	127	5	the	the	DET
cbi-57	127	6	sum	sum	NOUN
cbi-57	127	7	of	of	ADP
cbi-57	127	8	them	they	PRON
cbi-57	127	9	would	would	AUX
cbi-57	127	10	add	add	VERB
cbi-57	127	11	up	up	ADP
cbi-57	127	12	to	to	ADP
cbi-57	127	13	!	!	PUNCT
cbi-57	127	14	,	,	PUNCT
cbi-57	127	15	as	as	SCONJ
cbi-57	127	16	would	would	AUX
cbi-57	127	17	be	be	AUX
cbi-57	127	18	true	true	ADJ
cbi-57	127	19	for	for	ADP
cbi-57	127	20	any	any	DET
cbi-57	127	21	finite	finite	ADJ
cbi-57	127	22	value	value	NOUN
cbi-57	127	23	as	as	ADV
cbi-57	127	24	well	well	ADV
cbi-57	127	25	.	.	PUNCT
cbi-57	128	1	this	this	PRON
cbi-57	128	2	matches	match	VERB
cbi-57	128	3	the	the	DET
cbi-57	128	4	value	value	NOUN
cbi-57	128	5	given	give	VERB
cbi-57	128	6	by	by	ADP
cbi-57	128	7	equivalent	equivalent	ADJ
cbi-57	128	8	considerations	consideration	NOUN
cbi-57	128	9	in	in	ADP
cbi-57	128	10	(	(	PUNCT
cbi-57	128	11	1	1	NUM
cbi-57	128	12	)	)	PUNCT
cbi-57	128	13	.	.	PUNCT
cbi-57	129	1	the	the	DET
cbi-57	129	2	arithmetic	arithmetic	ADJ
cbi-57	129	3	series	series	NOUN
cbi-57	129	4	1	1	NUM
cbi-57	129	5	+	+	NUM
cbi-57	129	6	2	2	NUM
cbi-57	129	7	+	+	NUM
cbi-57	129	8	3	3	NUM
cbi-57	129	9	+	+	NUM
cbi-57	129	10	.	.	PUNCT
cbi-57	129	11	.	.	PUNCT
cbi-57	129	12	.	.	PUNCT
cbi-57	130	1	can	can	AUX
cbi-57	130	2	be	be	AUX
cbi-57	130	3	calculated	calculate	VERB
cbi-57	130	4	using	use	VERB
cbi-57	130	5	hyperreals	hyperreal	NOUN
cbi-57	130	6	as	as	ADV
cbi-57	130	7	well	well	ADV
cbi-57	130	8	.	.	PUNCT
cbi-57	130	9	!	!	PUNCT
cbi-57	130	10	!	!	PUNCT
cbi-57	131	1	i=1	i=1	PROPN
cbi-57	132	1	i	i	PRON
cbi-57	132	2	=	=	PUNCT
cbi-57	132	3	!	!	PUNCT
cbi-57	132	4	·	·	PUNCT
cbi-57	133	1	1	1	NUM
cbi-57	134	1	+	+	CCONJ
cbi-57	134	2	!	!	NUM
cbi-57	134	3	2	2	X
cbi-57	134	4	·	·	SYM
cbi-57	134	5	1	1	NUM
cbi-57	134	6	2	2	NUM
cbi-57	134	7	#	#	NOUN
cbi-57	134	8	!	!	PUNCT
cbi-57	134	9	·	·	PUNCT
cbi-57	135	1	1	1	NUM
cbi-57	135	2	2	2	NUM
cbi-57	135	3	(	(	PUNCT
cbi-57	135	4	11	11	NUM
cbi-57	135	5	)	)	PUNCT
cbi-57	135	6	=	=	PUNCT
cbi-57	135	7	!	!	SYM
cbi-57	135	8	2	2	NUM
cbi-57	135	9	2	2	NUM
cbi-57	135	10	+	+	CCONJ
cbi-57	135	11	!	!	PUNCT
cbi-57	135	12	2	2	X
cbi-57	135	13	.	.	PUNCT
cbi-57	136	1	(	(	PUNCT
cbi-57	136	2	12	12	NUM
cbi-57	136	3	)	)	PUNCT
cbi-57	136	4	the	the	DET
cbi-57	136	5	next	next	ADJ
cbi-57	136	6	arithmetic	arithmetic	ADJ
cbi-57	136	7	series	series	NOUN
cbi-57	136	8	to	to	PART
cbi-57	136	9	examine	examine	VERB
cbi-57	136	10	is	be	AUX
cbi-57	136	11	1	1	NUM
cbi-57	136	12	+	+	NOUN
cbi-57	136	13	3	3	NUM
cbi-57	136	14	+	+	NOUN
cbi-57	136	15	5	5	NUM
cbi-57	136	16	+	+	NUM
cbi-57	136	17	.	.	PUNCT
cbi-57	136	18	.	.	PUNCT
cbi-57	137	1	.	.	PUNCT
cbi-57	138	1	,	,	PUNCT
cbi-57	138	2	which	which	PRON
cbi-57	138	3	can	can	AUX
cbi-57	138	4	be	be	AUX
cbi-57	138	5	similarly	similarly	ADV
cbi-57	138	6	calculated	calculate	VERB
cbi-57	138	7	.	.	PUNCT
cbi-57	138	8	!	!	PUNCT
cbi-57	138	9	!	!	PUNCT
cbi-57	139	1	i=1	i=1	PROPN
cbi-57	139	2	(	(	PUNCT
cbi-57	139	3	2i	2i	NOUN
cbi-57	139	4	#	#	NOUN
cbi-57	139	5	1	1	NUM
cbi-57	139	6	)	)	PUNCT
cbi-57	139	7	=	=	PUNCT
cbi-57	139	8	!	!	PUNCT
cbi-57	139	9	·	·	PUNCT
cbi-57	140	1	1	1	NUM
cbi-57	141	1	+	+	CCONJ
cbi-57	141	2	!	!	NUM
cbi-57	141	3	2	2	X
cbi-57	141	4	·	·	SYM
cbi-57	141	5	2	2	NUM
cbi-57	141	6	2	2	NUM
cbi-57	141	7	#	#	NOUN
cbi-57	141	8	!	!	PUNCT
cbi-57	141	9	·	·	PUNCT
cbi-57	142	1	2	2	NUM
cbi-57	142	2	2	2	NUM
cbi-57	142	3	(	(	PUNCT
cbi-57	142	4	13	13	NUM
cbi-57	142	5	)	)	PUNCT
cbi-57	142	6	=	=	PUNCT
cbi-57	142	7	!	!	PUNCT
cbi-57	143	1	2	2	X
cbi-57	143	2	.	.	PUNCT
cbi-57	143	3	(	(	PUNCT
cbi-57	143	4	14	14	NUM
cbi-57	143	5	)	)	PUNCT
cbi-57	143	6	thus	thus	ADV
cbi-57	143	7	,	,	PUNCT
cbi-57	143	8	the	the	DET
cbi-57	143	9	value	value	NOUN
cbi-57	143	10	of	of	ADP
cbi-57	143	11	1	1	NUM
cbi-57	143	12	+	+	CCONJ
cbi-57	143	13	3	3	NUM
cbi-57	143	14	+	+	SYM
cbi-57	143	15	5	5	NUM
cbi-57	143	16	+	+	CCONJ
cbi-57	143	17	.	.	PUNCT
cbi-57	143	18	.	.	PUNCT
cbi-57	144	1	.	.	PUNCT
cbi-57	145	1	is	be	AUX
cbi-57	145	2	equal	equal	ADJ
cbi-57	145	3	to	to	ADP
cbi-57	145	4	(	(	PUNCT
cbi-57	145	5	1	1	NUM
cbi-57	145	6	+	+	SYM
cbi-57	145	7	1	1	NUM
cbi-57	145	8	+	+	SYM
cbi-57	145	9	1	1	NUM
cbi-57	145	10	.	.	PUNCT
cbi-57	145	11	.	.	PUNCT
cbi-57	146	1	.)2	.)2	PROPN
cbi-57	146	2	.	.	PUNCT
cbi-57	147	1	interestingly	interestingly	ADV
cbi-57	147	2	,	,	PUNCT
cbi-57	147	3	as	as	SCONJ
cbi-57	147	4	noted	note	VERB
cbi-57	147	5	in	in	ADP
cbi-57	147	6	section	section	NOUN
cbi-57	147	7	3	3	NUM
cbi-57	147	8	,	,	PUNCT
cbi-57	147	9	there	there	PRON
cbi-57	147	10	is	be	VERB
cbi-57	147	11	nothing	nothing	PRON
cbi-57	147	12	intrinsically	intrinsically	ADV
cbi-57	147	13	infinite	infinite	ADJ
cbi-57	147	14	about	about	ADP
cbi-57	147	15	the	the	DET
cbi-57	147	16	behavior	behavior	NOUN
cbi-57	147	17	of	of	ADP
cbi-57	147	18	!	!	PUNCT
cbi-57	148	1	in	in	ADP
cbi-57	148	2	these	these	DET
cbi-57	148	3	series	series	NOUN
cbi-57	148	4	.	.	PUNCT
cbi-57	149	1	for	for	ADP
cbi-57	149	2	instance	instance	NOUN
cbi-57	149	3	,	,	PUNCT
cbi-57	149	4	if	if	SCONJ
cbi-57	149	5	!	!	PUNCT
cbi-57	149	6	was	be	AUX
cbi-57	149	7	replaced	replace	VERB
cbi-57	149	8	with	with	ADP
cbi-57	149	9	5	5	NUM
cbi-57	149	10	,	,	PUNCT
cbi-57	149	11	the	the	DET
cbi-57	149	12	results	result	NOUN
cbi-57	149	13	would	would	AUX
cbi-57	149	14	hold	hold	VERB
cbi-57	149	15	.	.	PUNCT
cbi-57	150	1	that	that	PRON
cbi-57	150	2	is	is	ADV
cbi-57	150	3	,	,	PUNCT
cbi-57	150	4	(	(	PUNCT
cbi-57	150	5	1	1	NUM
cbi-57	150	6	+	+	SYM
cbi-57	150	7	1	1	NUM
cbi-57	150	8	+	+	NUM
cbi-57	150	9	1	1	NUM
cbi-57	150	10	+	+	NUM
cbi-57	150	11	1	1	NUM
cbi-57	151	1	+	+	X
cbi-57	151	2	1)2	1)2	NUM
cbi-57	151	3	=	=	SYM
cbi-57	151	4	(	(	PUNCT
cbi-57	151	5	1	1	NUM
cbi-57	151	6	+	+	NUM
cbi-57	151	7	3	3	NUM
cbi-57	151	8	+	+	SYM
cbi-57	151	9	5	5	NUM
cbi-57	151	10	+	+	CCONJ
cbi-57	151	11	7	7	NUM
cbi-57	151	12	+	+	CCONJ
cbi-57	151	13	9	9	NUM
cbi-57	151	14	)	)	PUNCT
cbi-57	151	15	=	=	SYM
cbi-57	152	1	25	25	NUM
cbi-57	152	2	.	.	PUNCT
cbi-57	153	1	even	even	ADV
cbi-57	153	2	though	though	SCONJ
cbi-57	153	3	the	the	DET
cbi-57	153	4	sums	sum	NOUN
cbi-57	153	5	are	be	AUX
cbi-57	153	6	divergent	divergent	ADJ
cbi-57	153	7	,	,	PUNCT
cbi-57	153	8	summing	sum	VERB
cbi-57	153	9	them	they	PRON
cbi-57	153	10	has	have	VERB
cbi-57	153	11	a	a	DET
cbi-57	153	12	very	very	ADV
cbi-57	153	13	well	well	ADV
cbi-57	153	14	-	-	PUNCT
cbi-57	153	15	defined	define	VERB
cbi-57	153	16	behavior	behavior	NOUN
cbi-57	153	17	within	within	ADP
cbi-57	153	18	the	the	DET
cbi-57	153	19	combined	combined	ADJ
cbi-57	153	20	hyperreal	hyperreal	NOUN
cbi-57	153	21	/	/	SYM
cbi-57	153	22	partial	partial	ADJ
cbi-57	153	23	sum	sum	NOUN
cbi-57	153	24	methodology	methodology	NOUN
cbi-57	153	25	presented	present	VERB
cbi-57	153	26	here	here	ADV
cbi-57	153	27	.	.	PUNCT
cbi-57	154	1	6.2	6.2	NUM
cbi-57	154	2	geometric	geometric	ADJ
cbi-57	154	3	series	series	NOUN
cbi-57	154	4	geometric	geometric	PROPN
cbi-57	154	5	series	series	NOUN
cbi-57	154	6	take	take	VERB
cbi-57	154	7	the	the	DET
cbi-57	154	8	form	form	NOUN
cbi-57	154	9	n	n	NOUN
cbi-57	154	10	!	!	PUNCT
cbi-57	155	1	i=1	i=1	PROPN
cbi-57	155	2	ar	ar	PROPN
cbi-57	155	3	i#1	i#1	PROPN
cbi-57	155	4	,	,	PUNCT
cbi-57	155	5	(	(	PUNCT
cbi-57	155	6	15	15	NUM
cbi-57	155	7	)	)	PUNCT
cbi-57	155	8	where	where	SCONJ
cbi-57	155	9	n	n	PRON
cbi-57	155	10	is	be	AUX
cbi-57	155	11	the	the	DET
cbi-57	155	12	number	number	NOUN
cbi-57	155	13	of	of	ADP
cbi-57	155	14	terms	term	NOUN
cbi-57	155	15	,	,	PUNCT
cbi-57	155	16	a	a	PRON
cbi-57	155	17	is	be	AUX
cbi-57	155	18	the	the	DET
cbi-57	155	19	starting	start	VERB
cbi-57	155	20	term	term	NOUN
cbi-57	155	21	,	,	PUNCT
cbi-57	155	22	and	and	CCONJ
cbi-57	155	23	r	r	NOUN
cbi-57	155	24	is	be	AUX
cbi-57	155	25	the	the	DET
cbi-57	155	26	common	common	ADJ
cbi-57	155	27	ratio	ratio	NOUN
cbi-57	155	28	.	.	PUNCT
cbi-57	156	1	a	a	DET
cbi-57	156	2	value	value	NOUN
cbi-57	156	3	for	for	ADP
cbi-57	156	4	a	a	DET
cbi-57	156	5	geometric	geometric	ADJ
cbi-57	156	6	series	series	NOUN
cbi-57	156	7	can	can	AUX
cbi-57	156	8	be	be	AUX
cbi-57	156	9	given	give	VERB
cbi-57	156	10	by	by	ADP
cbi-57	156	11	the	the	DET
cbi-57	156	12	formula	formula	NOUN
cbi-57	156	13	n	n	NOUN
cbi-57	156	14	!	!	PUNCT
cbi-57	157	1	i=1	i=1	PROPN
cbi-57	157	2	ar	ar	PROPN
cbi-57	157	3	i#1	i#1	PROPN
cbi-57	158	1	=	=	PUNCT
cbi-57	158	2	a	a	DET
cbi-57	158	3	1	1	NUM
cbi-57	158	4	#	#	NOUN
cbi-57	158	5	rn	rn	PROPN
cbi-57	158	6	1	1	NUM
cbi-57	158	7	#	#	NOUN
cbi-57	158	8	r	r	NOUN
cbi-57	158	9	.	.	PUNCT
cbi-57	159	1	(	(	PUNCT
cbi-57	159	2	16	16	NUM
cbi-57	159	3	)	)	PUNCT
cbi-57	159	4	because	because	SCONJ
cbi-57	159	5	an	an	DET
cbi-57	159	6	infinite	infinite	ADJ
cbi-57	159	7	series	series	NOUN
cbi-57	159	8	will	will	AUX
cbi-57	159	9	have	have	VERB
cbi-57	159	10	!	!	PUNCT
cbi-57	160	1	terms	term	NOUN
cbi-57	160	2	,	,	PUNCT
cbi-57	160	3	n	n	PRON
cbi-57	160	4	can	can	AUX
cbi-57	160	5	be	be	AUX
cbi-57	160	6	replaced	replace	VERB
cbi-57	160	7	with	with	ADP
cbi-57	160	8	!	!	PUNCT
cbi-57	160	9	.	.	PUNCT
cbi-57	161	1	let	let	VERB
cbi-57	161	2	us	we	PRON
cbi-57	161	3	begin	begin	VERB
cbi-57	161	4	by	by	ADP
cbi-57	161	5	looking	look	VERB
cbi-57	161	6	at	at	ADP
cbi-57	161	7	the	the	DET
cbi-57	161	8	series	series	NOUN
cbi-57	161	9	1	1	NUM
cbi-57	161	10	+	+	SYM
cbi-57	161	11	2	2	NUM
cbi-57	161	12	+	+	NUM
cbi-57	161	13	4	4	NUM
cbi-57	161	14	+	+	NUM
cbi-57	161	15	8	8	NUM
cbi-57	161	16	+	+	NUM
cbi-57	161	17	.	.	PUNCT
cbi-57	161	18	.	.	PUNCT
cbi-57	162	1	..	..	PUNCT
cbi-57	163	1	the	the	DET
cbi-57	163	2	value	value	NOUN
cbi-57	163	3	of	of	ADP
cbi-57	163	4	this	this	DET
cbi-57	163	5	series	series	NOUN
cbi-57	163	6	can	can	AUX
cbi-57	163	7	be	be	AUX
cbi-57	163	8	given	give	VERB
cbi-57	163	9	by	by	ADP
cbi-57	163	10	the	the	DET
cbi-57	163	11	formula	formula	NOUN
cbi-57	163	12	!	!	PUNCT
cbi-57	163	13	!	!	PUNCT
cbi-57	164	1	i=1	i=1	PROPN
cbi-57	165	1	2i#1	2i#1	NUM
cbi-57	165	2	=	=	SYM
cbi-57	165	3	1	1	NUM
cbi-57	165	4	·	·	SYM
cbi-57	165	5	1	1	NUM
cbi-57	165	6	#	#	NOUN
cbi-57	165	7	2	2	NUM
cbi-57	165	8	!	!	NOUN
cbi-57	165	9	1	1	NUM
cbi-57	165	10	#	#	NOUN
cbi-57	165	11	2	2	NUM
cbi-57	165	12	(	(	PUNCT
cbi-57	165	13	17	17	NUM
cbi-57	165	14	)	)	PUNCT
cbi-57	165	15	=	=	SYM
cbi-57	165	16	2	2	X
cbi-57	165	17	!	!	X
cbi-57	165	18	#	#	NOUN
cbi-57	165	19	1	1	NUM
cbi-57	165	20	.	.	PUNCT
cbi-57	166	1	(	(	PUNCT
cbi-57	166	2	18	18	NUM
cbi-57	166	3	)	)	SYM
cbi-57	166	4	10	10	NUM
cbi-57	166	5	hyperreal	hyperreal	ADJ
cbi-57	166	6	numbers	number	NOUN
cbi-57	166	7	for	for	ADP
cbi-57	166	8	infinite	infinite	ADJ
cbi-57	166	9	divergent	divergent	ADJ
cbi-57	166	10	series	series	NOUN
cbi-57	166	11	divergent	divergent	ADJ
cbi-57	166	12	geometric	geometric	ADJ
cbi-57	166	13	series	series	NOUN
cbi-57	166	14	will	will	AUX
cbi-57	166	15	generally	generally	ADV
cbi-57	166	16	have	have	VERB
cbi-57	166	17	the	the	DET
cbi-57	166	18	same	same	ADJ
cbi-57	166	19	form	form	NOUN
cbi-57	166	20	.	.	PUNCT
cbi-57	167	1	convergent	convergent	NOUN
cbi-57	167	2	series	serie	NOUN
cbi-57	167	3	are	be	AUX
cbi-57	167	4	also	also	ADV
cbi-57	167	5	interesting	interesting	ADJ
cbi-57	167	6	.	.	PUNCT
cbi-57	168	1	the	the	DET
cbi-57	168	2	series	series	NOUN
cbi-57	168	3	1	1	NUM
cbi-57	168	4	+	+	NUM
cbi-57	168	5	1	1	NUM
cbi-57	168	6	2	2	NUM
cbi-57	168	7	+	+	CCONJ
cbi-57	168	8	1	1	NUM
cbi-57	168	9	4	4	NUM
cbi-57	168	10	+	+	PUNCT
cbi-57	168	11	.	.	PUNCT
cbi-57	168	12	.	.	PUNCT
cbi-57	168	13	.	.	PUNCT
cbi-57	169	1	can	can	AUX
cbi-57	169	2	be	be	AUX
cbi-57	169	3	plugged	plug	VERB
cbi-57	169	4	into	into	ADP
cbi-57	169	5	the	the	DET
cbi-57	169	6	formula	formula	NOUN
cbi-57	169	7	to	to	PART
cbi-57	169	8	yield	yield	VERB
cbi-57	169	9	!	!	PUNCT
cbi-57	169	10	!	!	PUNCT
cbi-57	170	1	i=1	i=1	PRON
cbi-57	171	1	1	1	NUM
cbi-57	171	2	2	2	NUM
cbi-57	171	3	i#1	i#1	NOUN
cbi-57	171	4	=	=	SYM
cbi-57	171	5	1	1	NUM
cbi-57	171	6	·	·	SYM
cbi-57	171	7	1	1	NUM
cbi-57	171	8	#	#	NOUN
cbi-57	171	9	1	1	NUM
cbi-57	171	10	2	2	NUM
cbi-57	171	11	!	!	PUNCT
cbi-57	171	12	1	1	NUM
cbi-57	171	13	#	#	NOUN
cbi-57	171	14	1	1	NUM
cbi-57	171	15	2	2	NUM
cbi-57	171	16	(	(	PUNCT
cbi-57	171	17	19	19	NUM
cbi-57	171	18	)	)	PUNCT
cbi-57	171	19	=	=	SYM
cbi-57	171	20	2	2	NUM
cbi-57	171	21	#	#	NOUN
cbi-57	171	22	2	2	NUM
cbi-57	171	23	·	·	PUNCT
cbi-57	171	24	#	#	SYM
cbi-57	171	25	1	1	NUM
cbi-57	171	26	2	2	NUM
cbi-57	171	27	$	$	SYM
cbi-57	171	28	!	!	PUNCT
cbi-57	172	1	(	(	PUNCT
cbi-57	172	2	20	20	NUM
cbi-57	172	3	)	)	PUNCT
cbi-57	172	4	7	7	NUM
cbi-57	172	5	generalizing	generalize	VERB
cbi-57	172	6	to	to	ADP
cbi-57	172	7	the	the	DET
cbi-57	172	8	principal	principal	ADJ
cbi-57	172	9	value	value	NOUN
cbi-57	172	10	in	in	ADP
cbi-57	172	11	most	most	ADJ
cbi-57	172	12	discussions	discussion	NOUN
cbi-57	172	13	of	of	ADP
cbi-57	172	14	hyperreal	hyperreal	ADJ
cbi-57	172	15	numbers	number	NOUN
cbi-57	172	16	,	,	PUNCT
cbi-57	172	17	the	the	DET
cbi-57	172	18	halo	halo	NOUN
cbi-57	172	19	of	of	ADP
cbi-57	172	20	a	a	DET
cbi-57	172	21	number	number	NOUN
cbi-57	172	22	is	be	AUX
cbi-57	172	23	considered	consider	VERB
cbi-57	172	24	the	the	DET
cbi-57	172	25	hyperreal	hyperreal	ADJ
cbi-57	172	26	values	value	NOUN
cbi-57	172	27	which	which	PRON
cbi-57	172	28	are	be	AUX
cbi-57	172	29	infinitely	infinitely	ADV
cbi-57	172	30	close	close	ADJ
cbi-57	172	31	to	to	ADP
cbi-57	172	32	a	a	DET
cbi-57	172	33	standard	standard	ADJ
cbi-57	172	34	real	real	ADJ
cbi-57	172	35	number	number	NOUN
cbi-57	172	36	.	.	PUNCT
cbi-57	173	1	however	however	ADV
cbi-57	173	2	,	,	PUNCT
cbi-57	173	3	this	this	DET
cbi-57	173	4	definition	definition	NOUN
cbi-57	173	5	is	be	AUX
cbi-57	173	6	too	too	ADV
cbi-57	173	7	focused	focused	ADJ
cbi-57	173	8	on	on	ADP
cbi-57	173	9	real	real	ADJ
cbi-57	173	10	numbers	number	NOUN
cbi-57	173	11	.	.	PUNCT
cbi-57	174	1	we	we	PRON
cbi-57	174	2	will	will	AUX
cbi-57	174	3	consider	consider	VERB
cbi-57	174	4	the	the	DET
cbi-57	174	5	order	order	NOUN
cbi-57	174	6	of	of	ADP
cbi-57	174	7	a	a	DET
cbi-57	174	8	hyperreal	hyperreal	ADJ
cbi-57	174	9	value	value	NOUN
cbi-57	174	10	to	to	PART
cbi-57	174	11	be	be	AUX
cbi-57	174	12	its	its	PRON
cbi-57	174	13	largest	large	ADJ
cbi-57	174	14	exponent	exponent	NOUN
cbi-57	174	15	of	of	ADP
cbi-57	174	16	!	!	PUNCT
cbi-57	174	17	.	.	PUNCT
cbi-57	175	1	this	this	PRON
cbi-57	175	2	is	be	AUX
cbi-57	175	3	the	the	DET
cbi-57	175	4	most	most	ADV
cbi-57	175	5	significant	significant	ADJ
cbi-57	175	6	term	term	NOUN
cbi-57	175	7	of	of	ADP
cbi-57	175	8	the	the	DET
cbi-57	175	9	hyperreal	hyperreal	NOUN
cbi-57	175	10	value	value	NOUN
cbi-57	175	11	.	.	PUNCT
cbi-57	176	1	we	we	PRON
cbi-57	176	2	will	will	AUX
cbi-57	176	3	call	call	VERB
cbi-57	176	4	this	this	DET
cbi-57	176	5	most	most	ADV
cbi-57	176	6	significant	significant	ADJ
cbi-57	176	7	term	term	NOUN
cbi-57	176	8	the	the	DET
cbi-57	176	9	principal	principal	ADJ
cbi-57	176	10	value	value	NOUN
cbi-57	176	11	of	of	ADP
cbi-57	176	12	the	the	DET
cbi-57	176	13	hyperreal	hyperreal	NOUN
cbi-57	176	14	.	.	PUNCT
cbi-57	177	1	the	the	DET
cbi-57	177	2	halo	halo	NOUN
cbi-57	177	3	(	(	PUNCT
cbi-57	177	4	also	also	ADV
cbi-57	177	5	known	know	VERB
cbi-57	177	6	as	as	ADP
cbi-57	177	7	a	a	DET
cbi-57	177	8	monad	monad	PROPN
cbi-57	177	9	)	)	PUNCT
cbi-57	177	10	of	of	ADP
cbi-57	177	11	a	a	DET
cbi-57	177	12	hyperreal	hyperreal	NOUN
cbi-57	177	13	consists	consist	VERB
cbi-57	177	14	of	of	ADP
cbi-57	177	15	all	all	PRON
cbi-57	177	16	of	of	ADP
cbi-57	177	17	the	the	DET
cbi-57	177	18	hyperreals	hyperreal	NOUN
cbi-57	177	19	which	which	PRON
cbi-57	177	20	have	have	VERB
cbi-57	177	21	the	the	DET
cbi-57	177	22	same	same	ADJ
cbi-57	177	23	principal	principal	NOUN
cbi-57	177	24	value.4	value.4	PROPN
cbi-57	177	25	we	we	PRON
cbi-57	177	26	will	will	AUX
cbi-57	177	27	use	use	VERB
cbi-57	177	28	the	the	DET
cbi-57	177	29	$	$	SYM
cbi-57	177	30	operator	operator	NOUN
cbi-57	177	31	to	to	PART
cbi-57	177	32	denote	denote	VERB
cbi-57	177	33	two	two	NUM
cbi-57	177	34	hyperreals	hyperreal	NOUN
cbi-57	177	35	which	which	PRON
cbi-57	177	36	share	share	VERB
cbi-57	177	37	the	the	DET
cbi-57	177	38	same	same	ADJ
cbi-57	177	39	principal	principal	NOUN
cbi-57	178	1	value.5	value.5	INTJ
cbi-57	178	2	therefore	therefore	ADV
cbi-57	178	3	,	,	PUNCT
cbi-57	178	4	the	the	DET
cbi-57	178	5	halo	halo	NOUN
cbi-57	178	6	of	of	ADP
cbi-57	178	7	a	a	DET
cbi-57	178	8	hyperreal	hyperreal	NOUN
cbi-57	178	9	number	number	NOUN
cbi-57	178	10	consists	consist	VERB
cbi-57	178	11	of	of	ADP
cbi-57	178	12	all	all	PRON
cbi-57	178	13	of	of	ADP
cbi-57	178	14	those	those	DET
cbi-57	178	15	numbers	number	NOUN
cbi-57	178	16	which	which	PRON
cbi-57	178	17	share	share	VERB
cbi-57	178	18	the	the	DET
cbi-57	178	19	same	same	ADJ
cbi-57	178	20	principal	principal	ADJ
cbi-57	178	21	value	value	NOUN
cbi-57	178	22	.	.	PUNCT
cbi-57	179	1	many	many	ADJ
cbi-57	179	2	people	people	NOUN
cbi-57	179	3	use	use	VERB
cbi-57	179	4	“	"	PUNCT
cbi-57	179	5	infinitely	infinitely	ADV
cbi-57	179	6	close	close	ADJ
cbi-57	179	7	”	"	PUNCT
cbi-57	179	8	as	as	ADP
cbi-57	179	9	a	a	DET
cbi-57	179	10	colloquialism	colloquialism	NOUN
cbi-57	179	11	to	to	PART
cbi-57	179	12	describe	describe	VERB
cbi-57	179	13	two	two	NUM
cbi-57	179	14	hyperreals	hyperreal	NOUN
cbi-57	179	15	which	which	PRON
cbi-57	179	16	share	share	VERB
cbi-57	179	17	the	the	DET
cbi-57	179	18	same	same	ADJ
cbi-57	179	19	principal	principal	ADJ
cbi-57	179	20	value	value	NOUN
cbi-57	179	21	.	.	PUNCT
cbi-57	180	1	however	however	ADV
cbi-57	180	2	,	,	PUNCT
cbi-57	180	3	technically	technically	ADV
cbi-57	180	4	it	it	PRON
cbi-57	180	5	is	be	AUX
cbi-57	180	6	not	not	PART
cbi-57	180	7	correct	correct	ADJ
cbi-57	180	8	,	,	PUNCT
cbi-57	180	9	since	since	SCONJ
cbi-57	180	10	,	,	PUNCT
cbi-57	180	11	when	when	SCONJ
cbi-57	180	12	dealing	deal	VERB
cbi-57	180	13	4most	4most	NUM
cbi-57	180	14	texts	text	NOUN
cbi-57	180	15	on	on	ADP
cbi-57	180	16	hyperreal	hyperreal	ADJ
cbi-57	180	17	numbers	number	NOUN
cbi-57	180	18	define	define	VERB
cbi-57	180	19	the	the	DET
cbi-57	180	20	halo	halo	NOUN
cbi-57	180	21	or	or	CCONJ
cbi-57	180	22	monad	monad	PROPN
cbi-57	180	23	of	of	ADP
cbi-57	180	24	x	x	PRON
cbi-57	180	25	to	to	PART
cbi-57	180	26	be	be	AUX
cbi-57	180	27	all	all	PRON
cbi-57	180	28	of	of	ADP
cbi-57	180	29	the	the	DET
cbi-57	180	30	values	value	NOUN
cbi-57	180	31	y	y	PROPN
cbi-57	180	32	for	for	ADP
cbi-57	180	33	which	which	PRON
cbi-57	180	34	x	x	SYM
cbi-57	180	35	#	#	NOUN
cbi-57	180	36	y	y	PROPN
cbi-57	180	37	is	be	AUX
cbi-57	180	38	infinitesimal	infinitesimal	ADJ
cbi-57	180	39	(	(	PUNCT
cbi-57	180	40	loeb	loeb	NOUN
cbi-57	180	41	and	and	CCONJ
cbi-57	180	42	wol	wol	VERB
cbi-57	180	43	!	!	PUNCT
cbi-57	180	44	,	,	PUNCT
cbi-57	180	45	2015	2015	NUM
cbi-57	180	46	,	,	PUNCT
cbi-57	180	47	pg	pg	INTJ
cbi-57	180	48	.	.	PROPN
cbi-57	180	49	21	21	NUM
cbi-57	180	50	)	)	PUNCT
cbi-57	180	51	(	(	PUNCT
cbi-57	180	52	goldblatt	goldblatt	NOUN
cbi-57	180	53	,	,	PUNCT
cbi-57	180	54	1998	1998	NUM
cbi-57	180	55	,	,	PUNCT
cbi-57	180	56	pg	pg	INTJ
cbi-57	180	57	.	.	PROPN
cbi-57	180	58	52	52	NUM
cbi-57	180	59	)	)	PUNCT
cbi-57	180	60	.	.	PUNCT
cbi-57	181	1	however	however	ADV
cbi-57	181	2	,	,	PUNCT
cbi-57	181	3	defined	define	VERB
cbi-57	181	4	in	in	ADP
cbi-57	181	5	such	such	DET
cbi-57	181	6	a	a	DET
cbi-57	181	7	way	way	NOUN
cbi-57	181	8	,	,	PUNCT
cbi-57	181	9	the	the	DET
cbi-57	181	10	infinitesimals	infinitesimal	NOUN
cbi-57	181	11	!	!	PUNCT
cbi-57	182	1	#	#	SYM
cbi-57	182	2	1	1	NUM
cbi-57	182	3	and	and	CCONJ
cbi-57	182	4	2!#1	2!#1	NUM
cbi-57	182	5	are	be	AUX
cbi-57	182	6	within	within	ADP
cbi-57	182	7	a	a	DET
cbi-57	182	8	monad	monad	PROPN
cbi-57	182	9	.	.	PUNCT
cbi-57	183	1	using	use	VERB
cbi-57	183	2	principal	principal	ADJ
cbi-57	183	3	values	value	NOUN
cbi-57	183	4	,	,	PUNCT
cbi-57	183	5	!	!	PUNCT
cbi-57	184	1	#	#	SYM
cbi-57	184	2	1	1	NUM
cbi-57	184	3	and	and	CCONJ
cbi-57	184	4	2!#1	2!#1	NUM
cbi-57	184	5	are	be	AUX
cbi-57	184	6	in	in	ADP
cbi-57	184	7	the	the	DET
cbi-57	184	8	same	same	ADJ
cbi-57	184	9	galaxy	galaxy	NOUN
cbi-57	184	10	,	,	PUNCT
cbi-57	184	11	but	but	CCONJ
cbi-57	184	12	not	not	PART
cbi-57	184	13	the	the	DET
cbi-57	184	14	same	same	ADJ
cbi-57	184	15	monad	monad	PROPN
cbi-57	184	16	.	.	PUNCT
cbi-57	185	1	you	you	PRON
cbi-57	185	2	would	would	AUX
cbi-57	185	3	have	have	VERB
cbi-57	185	4	to	to	PART
cbi-57	185	5	have	have	VERB
cbi-57	185	6	a	a	DET
cbi-57	185	7	term	term	NOUN
cbi-57	185	8	of	of	ADP
cbi-57	185	9	lower	low	ADJ
cbi-57	185	10	-	-	PUNCT
cbi-57	185	11	order	order	NOUN
cbi-57	185	12	infinity	infinity	NOUN
cbi-57	185	13	to	to	PART
cbi-57	185	14	be	be	AUX
cbi-57	185	15	within	within	ADP
cbi-57	185	16	a	a	DET
cbi-57	185	17	monad	monad	PROPN
cbi-57	185	18	,	,	PUNCT
cbi-57	185	19	such	such	ADJ
cbi-57	185	20	as	as	ADP
cbi-57	185	21	!	!	PUNCT
cbi-57	185	22	#	#	SYM
cbi-57	185	23	1	1	NUM
cbi-57	185	24	and	and	CCONJ
cbi-57	185	25	!	!	PUNCT
cbi-57	186	1	#	#	SYM
cbi-57	186	2	1	1	NUM
cbi-57	186	3	+	+	CCONJ
cbi-57	186	4	!	!	PUNCT
cbi-57	186	5	#	#	SYM
cbi-57	186	6	2	2	NUM
cbi-57	186	7	.	.	PUNCT
cbi-57	187	1	this	this	PRON
cbi-57	187	2	seems	seem	VERB
cbi-57	187	3	to	to	PART
cbi-57	187	4	be	be	AUX
cbi-57	187	5	the	the	DET
cbi-57	187	6	essence	essence	NOUN
cbi-57	187	7	of	of	ADP
cbi-57	187	8	what	what	PRON
cbi-57	187	9	the	the	DET
cbi-57	187	10	other	other	ADJ
cbi-57	187	11	texts	text	NOUN
cbi-57	187	12	are	be	AUX
cbi-57	187	13	getting	get	VERB
cbi-57	187	14	at	at	ADP
cbi-57	187	15	,	,	PUNCT
cbi-57	187	16	but	but	CCONJ
cbi-57	187	17	,	,	PUNCT
cbi-57	187	18	since	since	SCONJ
cbi-57	187	19	most	most	ADJ
cbi-57	187	20	mathematics	mathematic	NOUN
cbi-57	187	21	focuses	focus	VERB
cbi-57	187	22	on	on	ADP
cbi-57	187	23	the	the	DET
cbi-57	187	24	reals	real	NOUN
cbi-57	187	25	,	,	PUNCT
cbi-57	187	26	their	their	PRON
cbi-57	187	27	definitions	definition	NOUN
cbi-57	187	28	were	be	AUX
cbi-57	187	29	entirely	entirely	ADV
cbi-57	187	30	based	base	VERB
cbi-57	187	31	on	on	ADP
cbi-57	187	32	using	use	VERB
cbi-57	187	33	reals	real	NOUN
cbi-57	187	34	as	as	ADP
cbi-57	187	35	a	a	DET
cbi-57	187	36	starting	starting	NOUN
cbi-57	187	37	point	point	NOUN
cbi-57	187	38	.	.	PUNCT
cbi-57	188	1	here	here	ADV
cbi-57	188	2	,	,	PUNCT
cbi-57	188	3	since	since	SCONJ
cbi-57	188	4	we	we	PRON
cbi-57	188	5	will	will	AUX
cbi-57	188	6	have	have	VERB
cbi-57	188	7	results	result	NOUN
cbi-57	188	8	in	in	ADP
cbi-57	188	9	the	the	DET
cbi-57	188	10	hyperreals	hyperreal	NOUN
cbi-57	188	11	,	,	PUNCT
cbi-57	188	12	we	we	PRON
cbi-57	188	13	need	need	VERB
cbi-57	188	14	definitions	definition	NOUN
cbi-57	188	15	that	that	PRON
cbi-57	188	16	are	be	AUX
cbi-57	188	17	equally	equally	ADV
cbi-57	188	18	useful	useful	ADJ
cbi-57	188	19	when	when	SCONJ
cbi-57	188	20	the	the	DET
cbi-57	188	21	final	final	ADJ
cbi-57	188	22	result	result	NOUN
cbi-57	188	23	is	be	AUX
cbi-57	188	24	a	a	DET
cbi-57	188	25	hyperreal	hyperreal	ADJ
cbi-57	188	26	number	number	NOUN
cbi-57	188	27	.	.	PUNCT
cbi-57	189	1	5	5	NUM
cbi-57	189	2	in	in	ADP
cbi-57	189	3	practice	practice	NOUN
cbi-57	189	4	,	,	PUNCT
cbi-57	189	5	$	$	PRON
cbi-57	189	6	can	can	AUX
cbi-57	189	7	be	be	AUX
cbi-57	189	8	replaced	replace	VERB
cbi-57	189	9	with	with	ADP
cbi-57	189	10	=	=	NOUN
cbi-57	189	11	,	,	PUNCT
cbi-57	189	12	as	as	SCONJ
cbi-57	189	13	it	it	PRON
cbi-57	189	14	denotes	denote	VERB
cbi-57	189	15	equality	equality	NOUN
cbi-57	189	16	to	to	ADP
cbi-57	189	17	the	the	DET
cbi-57	189	18	extent	extent	NOUN
cbi-57	189	19	normally	normally	ADV
cbi-57	189	20	practiced	practice	VERB
cbi-57	189	21	in	in	ADP
cbi-57	189	22	mathematics	mathematic	NOUN
cbi-57	189	23	.	.	PUNCT
cbi-57	190	1	for	for	ADP
cbi-57	190	2	instance	instance	NOUN
cbi-57	190	3	,	,	PUNCT
cbi-57	190	4	the	the	DET
cbi-57	190	5	di!erential	di!erential	ADJ
cbi-57	190	6	d	d	X
cbi-57	190	7	(	(	PUNCT
cbi-57	190	8	xy	xy	NOUN
cbi-57	190	9	)	)	PUNCT
cbi-57	190	10	is	be	AUX
cbi-57	190	11	often	often	ADV
cbi-57	190	12	stated	state	VERB
cbi-57	190	13	as	as	ADP
cbi-57	190	14	being	be	AUX
cbi-57	190	15	equal	equal	ADJ
cbi-57	190	16	to	to	ADP
cbi-57	190	17	x	x	PROPN
cbi-57	190	18	dy	dy	NOUN
cbi-57	190	19	+	+	X
cbi-57	190	20	y	y	PROPN
cbi-57	190	21	dx	dx	PROPN
cbi-57	190	22	,	,	PUNCT
cbi-57	190	23	but	but	CCONJ
cbi-57	190	24	really	really	ADV
cbi-57	190	25	it	it	PRON
cbi-57	190	26	is	be	AUX
cbi-57	190	27	just	just	ADV
cbi-57	190	28	the	the	DET
cbi-57	190	29	principal	principal	ADJ
cbi-57	190	30	value	value	NOUN
cbi-57	190	31	.	.	PUNCT
cbi-57	191	1	the	the	DET
cbi-57	191	2	actual	actual	ADJ
cbi-57	191	3	value	value	NOUN
cbi-57	191	4	is	be	AUX
cbi-57	191	5	x	x	PART
cbi-57	191	6	dy	dy	NOUN
cbi-57	192	1	+	+	CCONJ
cbi-57	192	2	y	y	PROPN
cbi-57	192	3	dx	dx	PROPN
cbi-57	192	4	+	+	CCONJ
cbi-57	192	5	dy	dy	PROPN
cbi-57	192	6	dx	dx	PROPN
cbi-57	192	7	.	.	PUNCT
cbi-57	193	1	the	the	DET
cbi-57	193	2	dy	dy	PROPN
cbi-57	193	3	dx	dx	PROPN
cbi-57	193	4	term	term	NOUN
cbi-57	193	5	is	be	AUX
cbi-57	193	6	always	always	ADV
cbi-57	193	7	discarded	discard	VERB
cbi-57	193	8	because	because	SCONJ
cbi-57	193	9	it	it	PRON
cbi-57	193	10	is	be	AUX
cbi-57	193	11	infinitely	infinitely	ADV
cbi-57	193	12	less	less	ADV
cbi-57	193	13	significant	significant	ADJ
cbi-57	193	14	than	than	ADP
cbi-57	193	15	the	the	DET
cbi-57	193	16	other	other	ADJ
cbi-57	193	17	pieces	piece	NOUN
cbi-57	193	18	.	.	PUNCT
cbi-57	194	1	even	even	ADV
cbi-57	194	2	when	when	SCONJ
cbi-57	194	3	discarding	discard	VERB
cbi-57	194	4	this	this	DET
cbi-57	194	5	term	term	NOUN
cbi-57	194	6	,	,	PUNCT
cbi-57	194	7	the	the	DET
cbi-57	194	8	equality	equality	NOUN
cbi-57	194	9	sign	sign	NOUN
cbi-57	194	10	is	be	AUX
cbi-57	194	11	used	use	VERB
cbi-57	194	12	.	.	PUNCT
cbi-57	195	1	therefore	therefore	ADV
cbi-57	195	2	,	,	PUNCT
cbi-57	195	3	while	while	SCONJ
cbi-57	195	4	the	the	DET
cbi-57	195	5	present	present	ADJ
cbi-57	195	6	paper	paper	NOUN
cbi-57	195	7	will	will	AUX
cbi-57	195	8	be	be	AUX
cbi-57	195	9	pedantic	pedantic	ADJ
cbi-57	195	10	about	about	ADP
cbi-57	195	11	asserting	assert	VERB
cbi-57	195	12	exact	exact	ADJ
cbi-57	195	13	equality	equality	NOUN
cbi-57	195	14	or	or	CCONJ
cbi-57	195	15	mere	mere	ADJ
cbi-57	195	16	principal	principal	ADJ
cbi-57	195	17	value	value	NOUN
cbi-57	195	18	,	,	PUNCT
cbi-57	195	19	for	for	SCONJ
cbi-57	195	20	most	most	ADJ
cbi-57	195	21	general	general	ADJ
cbi-57	195	22	purposes	purpose	NOUN
cbi-57	195	23	equality	equality	NOUN
cbi-57	195	24	can	can	AUX
cbi-57	195	25	be	be	AUX
cbi-57	195	26	asserted	assert	VERB
cbi-57	195	27	even	even	ADV
cbi-57	195	28	when	when	SCONJ
cbi-57	195	29	only	only	ADV
cbi-57	195	30	stating	state	VERB
cbi-57	195	31	the	the	DET
cbi-57	195	32	principal	principal	ADJ
cbi-57	195	33	value	value	NOUN
cbi-57	195	34	.	.	PUNCT
cbi-57	196	1	with	with	ADP
cbi-57	196	2	infinities	infinity	NOUN
cbi-57	196	3	,	,	PUNCT
cbi-57	196	4	two	two	NUM
cbi-57	196	5	hyperreals	hyperreal	NOUN
cbi-57	196	6	which	which	PRON
cbi-57	196	7	di!er	di!er	VERB
cbi-57	196	8	by	by	ADP
cbi-57	196	9	multiple	multiple	ADJ
cbi-57	196	10	infinities	infinity	NOUN
cbi-57	196	11	can	can	AUX
cbi-57	196	12	be	be	AUX
cbi-57	196	13	considered	consider	VERB
cbi-57	196	14	“	"	PUNCT
cbi-57	196	15	infinitely	infinitely	ADV
cbi-57	196	16	close	close	ADJ
cbi-57	196	17	.	.	PUNCT
cbi-57	196	18	”	"	PUNCT
cbi-57	197	1	that	that	PRON
cbi-57	197	2	is	be	AUX
cbi-57	197	3	,	,	PUNCT
cbi-57	197	4	!	!	PUNCT
cbi-57	198	1	2	2	NUM
cbi-57	198	2	+	+	NOUN
cbi-57	198	3	5	5	NUM
cbi-57	198	4	!	!	NUM
cbi-57	198	5	,	,	PUNCT
cbi-57	198	6	!	!	PUNCT
cbi-57	199	1	2	2	NUM
cbi-57	199	2	#	#	NOUN
cbi-57	199	3	12	12	NUM
cbi-57	199	4	!	!	PUNCT
cbi-57	199	5	,	,	PUNCT
cbi-57	199	6	and	and	CCONJ
cbi-57	199	7	!	!	PUNCT
cbi-57	199	8	2	2	NUM
cbi-57	199	9	+	+	SYM
cbi-57	199	10	23	23	NUM
cbi-57	199	11	all	all	PRON
cbi-57	199	12	share	share	VERB
cbi-57	199	13	the	the	DET
cbi-57	199	14	same	same	ADJ
cbi-57	199	15	principal	principal	ADJ
cbi-57	199	16	value	value	NOUN
cbi-57	199	17	,	,	PUNCT
cbi-57	199	18	!	!	PUNCT
cbi-57	200	1	2	2	X
cbi-57	200	2	.	.	X
cbi-57	200	3	they	they	PRON
cbi-57	200	4	are	be	AUX
cbi-57	200	5	infinitely	infinitely	ADV
cbi-57	200	6	apart	apart	ADV
cbi-57	200	7	,	,	PUNCT
cbi-57	200	8	yet	yet	ADV
cbi-57	200	9	,	,	PUNCT
cbi-57	200	10	colloquially	colloquially	ADV
cbi-57	200	11	,	,	PUNCT
cbi-57	200	12	they	they	PRON
cbi-57	200	13	can	can	AUX
cbi-57	200	14	be	be	AUX
cbi-57	200	15	considered	consider	VERB
cbi-57	200	16	“	"	PUNCT
cbi-57	200	17	infinitely	infinitely	ADV
cbi-57	200	18	close	close	ADJ
cbi-57	200	19	”	"	PUNCT
cbi-57	200	20	because	because	SCONJ
cbi-57	200	21	their	their	PRON
cbi-57	200	22	di!erences	di!erence	NOUN
cbi-57	200	23	are	be	AUX
cbi-57	200	24	infinitely	infinitely	ADV
cbi-57	200	25	less	less	ADV
cbi-57	200	26	significant	significant	ADJ
cbi-57	200	27	than	than	ADP
cbi-57	200	28	their	their	PRON
cbi-57	200	29	similarities	similarity	NOUN
cbi-57	200	30	.	.	PUNCT
cbi-57	201	1	when	when	SCONJ
cbi-57	201	2	dealing	deal	VERB
cbi-57	201	3	with	with	ADP
cbi-57	201	4	hyperreals	hyperreal	NOUN
cbi-57	201	5	,	,	PUNCT
cbi-57	201	6	the	the	DET
cbi-57	201	7	principal	principal	ADJ
cbi-57	201	8	value	value	NOUN
cbi-57	201	9	is	be	AUX
cbi-57	201	10	the	the	DET
cbi-57	201	11	main	main	ADJ
cbi-57	201	12	one	one	NUM
cbi-57	201	13	of	of	ADP
cbi-57	201	14	concern	concern	NOUN
cbi-57	201	15	.	.	PUNCT
cbi-57	202	1	so	so	ADV
cbi-57	202	2	,	,	PUNCT
cbi-57	202	3	for	for	ADP
cbi-57	202	4	instance	instance	NOUN
cbi-57	202	5	,	,	PUNCT
cbi-57	202	6	while	while	SCONJ
cbi-57	202	7	1	1	NUM
cbi-57	202	8	+	+	SYM
cbi-57	202	9	2	2	NUM
cbi-57	202	10	+	+	NUM
cbi-57	202	11	3	3	NUM
cbi-57	202	12	+	+	NUM
cbi-57	202	13	.	.	PUNCT
cbi-57	202	14	.	.	PUNCT
cbi-57	202	15	.	.	PUNCT
cbi-57	202	16	is	be	AUX
cbi-57	202	17	exactly	exactly	ADV
cbi-57	202	18	described	describe	VERB
cbi-57	202	19	by	by	ADP
cbi-57	202	20	!	!	PROPN
cbi-57	202	21	2	2	NUM
cbi-57	202	22	2	2	NUM
cbi-57	202	23	+	+	CCONJ
cbi-57	202	24	!	!	PUNCT
cbi-57	202	25	2	2	NUM
cbi-57	202	26	,	,	PUNCT
cbi-57	202	27	its	its	PRON
cbi-57	202	28	principal	principal	ADJ
cbi-57	202	29	value	value	NOUN
cbi-57	202	30	is	be	AUX
cbi-57	202	31	just	just	ADV
cbi-57	202	32	!	!	PUNCT
cbi-57	202	33	2	2	NUM
cbi-57	202	34	2	2	NUM
cbi-57	202	35	.	.	PUNCT
cbi-57	203	1	therefore	therefore	ADV
cbi-57	203	2	,	,	PUNCT
cbi-57	203	3	the	the	DET
cbi-57	203	4	formula	formula	NOUN
cbi-57	203	5	given	give	VERB
cbi-57	203	6	in	in	ADP
cbi-57	203	7	(	(	PUNCT
cbi-57	203	8	7	7	NUM
cbi-57	203	9	)	)	PUNCT
cbi-57	203	10	can	can	AUX
cbi-57	203	11	actually	actually	ADV
cbi-57	203	12	be	be	AUX
cbi-57	203	13	simplified	simplify	VERB
cbi-57	203	14	to	to	ADP
cbi-57	203	15	!	!	PUNCT
cbi-57	203	16	!	!	PUNCT
cbi-57	204	1	i=1	i=1	PROPN
cbi-57	205	1	a	a	PRON
cbi-57	206	1	+	+	NOUN
cbi-57	206	2	(	(	PUNCT
cbi-57	206	3	i	i	NOUN
cbi-57	206	4	#	#	NOUN
cbi-57	206	5	1)d	1)d	NUM
cbi-57	206	6	$	$	SYM
cbi-57	206	7	!	!	PUNCT
cbi-57	207	1	2d	2d	NOUN
cbi-57	207	2	2	2	NUM
cbi-57	207	3	(	(	PUNCT
cbi-57	207	4	21	21	NUM
cbi-57	207	5	)	)	PUNCT
cbi-57	207	6	if	if	SCONJ
cbi-57	207	7	d	d	PROPN
cbi-57	207	8	!	!	PUNCT
cbi-57	208	1	0.6	0.6	NUM
cbi-57	208	2	interestingly	interestingly	ADV
cbi-57	208	3	,	,	PUNCT
cbi-57	208	4	we	we	PRON
cbi-57	208	5	can	can	AUX
cbi-57	208	6	see	see	VERB
cbi-57	208	7	that	that	PRON
cbi-57	208	8	,	,	PUNCT
cbi-57	208	9	while	while	SCONJ
cbi-57	208	10	the	the	DET
cbi-57	208	11	exact	exact	ADJ
cbi-57	208	12	value	value	NOUN
cbi-57	208	13	of	of	ADP
cbi-57	208	14	the	the	DET
cbi-57	208	15	hyperreal	hyperreal	NOUN
cbi-57	208	16	associated	associate	VERB
cbi-57	208	17	with	with	ADP
cbi-57	208	18	a	a	DET
cbi-57	208	19	series	series	NOUN
cbi-57	208	20	depends	depend	VERB
cbi-57	208	21	on	on	ADP
cbi-57	208	22	the	the	DET
cbi-57	208	23	starting	starting	NOUN
cbi-57	208	24	point	point	NOUN
cbi-57	208	25	,	,	PUNCT
cbi-57	208	26	the	the	DET
cbi-57	208	27	principal	principal	ADJ
cbi-57	208	28	value	value	NOUN
cbi-57	208	29	depends	depend	VERB
cbi-57	208	30	only	only	ADV
cbi-57	208	31	on	on	ADP
cbi-57	208	32	the	the	DET
cbi-57	208	33	distance	distance	NOUN
cbi-57	208	34	chosen	choose	VERB
cbi-57	208	35	,	,	PUNCT
cbi-57	208	36	provided	provide	VERB
cbi-57	208	37	that	that	PRON
cbi-57	208	38	d	d	NOUN
cbi-57	208	39	!	!	PUNCT
cbi-57	208	40	0	0	X
cbi-57	208	41	.	.	PUNCT
cbi-57	209	1	geometric	geometric	ADJ
cbi-57	209	2	series	series	NOUN
cbi-57	209	3	can	can	AUX
cbi-57	209	4	use	use	VERB
cbi-57	209	5	similar	similar	ADJ
cbi-57	209	6	considerations	consideration	NOUN
cbi-57	209	7	.	.	PUNCT
cbi-57	210	1	you	you	PRON
cbi-57	210	2	may	may	AUX
cbi-57	210	3	have	have	AUX
cbi-57	210	4	noticed	notice	VERB
cbi-57	210	5	that	that	SCONJ
cbi-57	210	6	the	the	DET
cbi-57	210	7	hyperreal	hyperreal	NOUN
cbi-57	210	8	given	give	VERB
cbi-57	210	9	for	for	ADP
cbi-57	210	10	the	the	DET
cbi-57	210	11	series	series	NOUN
cbi-57	210	12	1	1	NUM
cbi-57	210	13	+	+	CCONJ
cbi-57	210	14	1	1	NUM
cbi-57	210	15	2	2	NUM
cbi-57	210	16	+	+	CCONJ
cbi-57	210	17	1	1	NUM
cbi-57	210	18	4	4	NUM
cbi-57	210	19	+	+	PUNCT
cbi-57	210	20	.	.	PUNCT
cbi-57	210	21	.	.	PUNCT
cbi-57	210	22	.	.	PUNCT
cbi-57	211	1	in	in	ADP
cbi-57	211	2	section	section	NOUN
cbi-57	211	3	6	6	NUM
cbi-57	211	4	is	be	AUX
cbi-57	211	5	2	2	NUM
cbi-57	211	6	#	#	SYM
cbi-57	211	7	2	2	NUM
cbi-57	211	8	·	·	PUNCT
cbi-57	211	9	%	%	NOUN
cbi-57	211	10	1	1	NUM
cbi-57	211	11	2	2	NUM
cbi-57	211	12	&	&	CCONJ
cbi-57	211	13	!	!	PUNCT
cbi-57	211	14	.	.	PUNCT
cbi-57	212	1	typically	typically	ADV
cbi-57	212	2	,	,	PUNCT
cbi-57	212	3	this	this	DET
cbi-57	212	4	series	series	NOUN
cbi-57	212	5	is	be	AUX
cbi-57	212	6	thought	think	VERB
cbi-57	212	7	to	to	PART
cbi-57	212	8	converge	converge	VERB
cbi-57	212	9	to	to	ADP
cbi-57	212	10	2	2	NUM
cbi-57	212	11	.	.	PUNCT
cbi-57	213	1	in	in	ADP
cbi-57	213	2	fact	fact	NOUN
cbi-57	213	3	,	,	PUNCT
cbi-57	213	4	its	its	PRON
cbi-57	213	5	principal	principal	ADJ
cbi-57	213	6	value	value	NOUN
cbi-57	213	7	is	be	AUX
cbi-57	213	8	2	2	NUM
cbi-57	213	9	,	,	PUNCT
cbi-57	213	10	because	because	SCONJ
cbi-57	213	11	%	%	NOUN
cbi-57	213	12	1	1	NUM
cbi-57	213	13	2	2	NUM
cbi-57	213	14	&	&	CCONJ
cbi-57	213	15	!	!	PROPN
cbi-57	213	16	is	be	AUX
cbi-57	213	17	an	an	DET
cbi-57	213	18	infinitesimal	infinitesimal	NOUN
cbi-57	213	19	.	.	PUNCT
cbi-57	214	1	the	the	DET
cbi-57	214	2	use	use	NOUN
cbi-57	214	3	of	of	ADP
cbi-57	214	4	principal	principal	ADJ
cbi-57	214	5	values	value	NOUN
cbi-57	214	6	allows	allow	VERB
cbi-57	214	7	for	for	ADP
cbi-57	214	8	a	a	DET
cbi-57	214	9	great	great	ADJ
cbi-57	214	10	amount	amount	NOUN
cbi-57	214	11	of	of	ADP
cbi-57	214	12	simplification	simplification	NOUN
cbi-57	214	13	for	for	ADP
cbi-57	214	14	hyperreal	hyperreal	ADJ
cbi-57	214	15	values	value	NOUN
cbi-57	214	16	and	and	CCONJ
cbi-57	214	17	formulas	formula	NOUN
cbi-57	214	18	.	.	PUNCT
cbi-57	215	1	as	as	ADP
cbi-57	215	2	an	an	DET
cbi-57	215	3	example	example	NOUN
cbi-57	215	4	,	,	PUNCT
cbi-57	215	5	the	the	DET
cbi-57	215	6	ratio	ratio	NOUN
cbi-57	215	7	between	between	ADP
cbi-57	215	8	two	two	NUM
cbi-57	215	9	given	give	VERB
cbi-57	215	10	arithmetic	arithmetic	ADJ
cbi-57	215	11	series	series	NOUN
cbi-57	215	12	can	can	AUX
cbi-57	215	13	be	be	AUX
cbi-57	215	14	solved	solve	VERB
cbi-57	215	15	for	for	ADP
cbi-57	215	16	very	very	ADV
cbi-57	215	17	simply	simply	ADV
cbi-57	215	18	.	.	PUNCT
cbi-57	216	1	s1	s1	NOUN
cbi-57	216	2	=	=	PUNCT
cbi-57	216	3	!	!	PUNCT
cbi-57	216	4	!	!	PUNCT
cbi-57	217	1	i=1	i=1	PROPN
cbi-57	217	2	a1	a1	NOUN
cbi-57	218	1	+	+	CCONJ
cbi-57	218	2	(	(	PUNCT
cbi-57	219	1	i	i	NOUN
cbi-57	219	2	#	#	NOUN
cbi-57	219	3	1)d1	1)d1	NUM
cbi-57	219	4	$	$	SYM
cbi-57	219	5	!	!	SYM
cbi-57	219	6	2(d1	2(d1	NUM
cbi-57	219	7	)	)	PUNCT
cbi-57	219	8	2	2	NUM
cbi-57	219	9	s2	s2	NOUN
cbi-57	219	10	=	=	PUNCT
cbi-57	219	11	!	!	PUNCT
cbi-57	219	12	!	!	PUNCT
cbi-57	220	1	i=1	i=1	PROPN
cbi-57	220	2	a2	a2	PROPN
cbi-57	221	1	+	+	CCONJ
cbi-57	222	1	(	(	PUNCT
cbi-57	222	2	i	i	PRON
cbi-57	222	3	#	#	NOUN
cbi-57	222	4	1)d2	1)d2	NUM
cbi-57	222	5	$	$	SYM
cbi-57	222	6	!	!	PUNCT
cbi-57	222	7	2(d2	2(d2	NUM
cbi-57	222	8	)	)	PUNCT
cbi-57	222	9	2	2	NUM
cbi-57	222	10	s1	s1	NOUN
cbi-57	222	11	s2	s2	X
cbi-57	222	12	$	$	SYM
cbi-57	222	13	!	!	SYM
cbi-57	222	14	2	2	NUM
cbi-57	222	15	(	(	PUNCT
cbi-57	222	16	d1	d1	NOUN
cbi-57	222	17	)	)	PUNCT
cbi-57	222	18	2	2	NUM
cbi-57	222	19	!	!	SYM
cbi-57	222	20	2	2	NUM
cbi-57	222	21	(	(	PUNCT
cbi-57	222	22	d2	d2	NOUN
cbi-57	222	23	)	)	PUNCT
cbi-57	222	24	2	2	NUM
cbi-57	222	25	=	=	SYM
cbi-57	222	26	d1	d1	PROPN
cbi-57	222	27	d2	d2	PROPN
cbi-57	222	28	(	(	PUNCT
cbi-57	222	29	22	22	NUM
cbi-57	222	30	)	)	PUNCT
cbi-57	222	31	in	in	ADP
cbi-57	222	32	other	other	ADJ
cbi-57	222	33	words	word	NOUN
cbi-57	222	34	,	,	PUNCT
cbi-57	222	35	the	the	DET
cbi-57	222	36	principal	principal	ADJ
cbi-57	222	37	value	value	NOUN
cbi-57	222	38	of	of	ADP
cbi-57	222	39	the	the	DET
cbi-57	222	40	ratio	ratio	NOUN
cbi-57	222	41	of	of	ADP
cbi-57	222	42	two	two	NUM
cbi-57	222	43	arithmetic	arithmetic	ADJ
cbi-57	222	44	series	series	NOUN
cbi-57	222	45	is	be	AUX
cbi-57	222	46	simply	simply	ADV
cbi-57	222	47	the	the	DET
cbi-57	222	48	ratio	ratio	NOUN
cbi-57	222	49	of	of	ADP
cbi-57	222	50	the	the	DET
cbi-57	222	51	distances	distance	NOUN
cbi-57	222	52	.	.	PUNCT
cbi-57	223	1	6when	6when	NUM
cbi-57	223	2	d	d	NOUN
cbi-57	223	3	=	=	SYM
cbi-57	223	4	0	0	NUM
cbi-57	223	5	,	,	PUNCT
cbi-57	223	6	then	then	ADV
cbi-57	223	7	the	the	DET
cbi-57	223	8	!	!	SYM
cbi-57	223	9	2	2	NUM
cbi-57	223	10	term	term	NOUN
cbi-57	223	11	goes	go	VERB
cbi-57	223	12	to	to	ADP
cbi-57	223	13	zero	zero	NUM
cbi-57	223	14	,	,	PUNCT
cbi-57	223	15	and	and	CCONJ
cbi-57	223	16	the	the	DET
cbi-57	223	17	series	series	NOUN
cbi-57	223	18	simplifies	simplifie	NOUN
cbi-57	223	19	to	to	ADP
cbi-57	223	20	a	a	PRON
cbi-57	223	21	·	·	PUNCT
cbi-57	223	22	!	!	PUNCT
cbi-57	224	1	instead	instead	ADV
cbi-57	224	2	.	.	PUNCT
cbi-57	225	1	volume	volume	NOUN
cbi-57	225	2	2	2	NUM
cbi-57	225	3	,	,	PUNCT
cbi-57	225	4	issue	issue	NOUN
cbi-57	225	5	1	1	NUM
cbi-57	225	6	8	8	NUM
cbi-57	225	7	.	.	PUNCT
cbi-57	226	1	series	series	NOUN
cbi-57	226	2	manipulation	manipulation	NOUN
cbi-57	226	3	rules	rule	VERB
cbi-57	226	4	for	for	ADP
cbi-57	226	5	finite	finite	ADJ
cbi-57	226	6	subsets	subset	NOUN
cbi-57	226	7	11	11	NUM
cbi-57	226	8	8	8	NUM
cbi-57	226	9	series	series	NOUN
cbi-57	226	10	manipulation	manipulation	NOUN
cbi-57	226	11	rules	rule	NOUN
cbi-57	226	12	for	for	ADP
cbi-57	226	13	finite	finite	ADJ
cbi-57	226	14	subsets	subset	NOUN
cbi-57	226	15	many	many	ADJ
cbi-57	226	16	attempts	attempt	NOUN
cbi-57	226	17	to	to	PART
cbi-57	226	18	manipulate	manipulate	VERB
cbi-57	226	19	divergent	divergent	ADJ
cbi-57	226	20	series	series	NOUN
cbi-57	226	21	have	have	AUX
cbi-57	226	22	resulted	result	VERB
cbi-57	226	23	in	in	ADP
cbi-57	226	24	contradictions	contradiction	NOUN
cbi-57	226	25	,	,	PUNCT
cbi-57	226	26	to	to	ADP
cbi-57	226	27	the	the	DET
cbi-57	226	28	extent	extent	NOUN
cbi-57	226	29	that	that	SCONJ
cbi-57	226	30	many	many	ADJ
cbi-57	226	31	suggest	suggest	VERB
cbi-57	226	32	that	that	SCONJ
cbi-57	226	33	it	it	PRON
cbi-57	226	34	is	be	AUX
cbi-57	226	35	best	good	ADJ
cbi-57	226	36	to	to	PART
cbi-57	226	37	not	not	PART
cbi-57	226	38	attempt	attempt	VERB
cbi-57	226	39	to	to	PART
cbi-57	226	40	do	do	VERB
cbi-57	226	41	so	so	ADV
cbi-57	226	42	.	.	PUNCT
cbi-57	227	1	the	the	DET
cbi-57	227	2	reason	reason	NOUN
cbi-57	227	3	for	for	ADP
cbi-57	227	4	these	these	DET
cbi-57	227	5	contradictions	contradiction	NOUN
cbi-57	227	6	,	,	PUNCT
cbi-57	227	7	however	however	ADV
cbi-57	227	8	,	,	PUNCT
cbi-57	227	9	lies	lie	VERB
cbi-57	227	10	in	in	ADP
cbi-57	227	11	the	the	DET
cbi-57	227	12	treatment	treatment	NOUN
cbi-57	227	13	of	of	ADP
cbi-57	227	14	the	the	DET
cbi-57	227	15	infinite	infinite	ADJ
cbi-57	227	16	nature	nature	NOUN
cbi-57	227	17	of	of	ADP
cbi-57	227	18	the	the	DET
cbi-57	227	19	number	number	NOUN
cbi-57	227	20	of	of	ADP
cbi-57	227	21	values	value	NOUN
cbi-57	227	22	.	.	PUNCT
cbi-57	228	1	in	in	ADP
cbi-57	228	2	the	the	DET
cbi-57	228	3	real	real	ADJ
cbi-57	228	4	system	system	NOUN
cbi-57	228	5	,	,	PUNCT
cbi-57	228	6	"	"	PUNCT
cbi-57	228	7	is	be	AUX
cbi-57	228	8	considered	consider	VERB
cbi-57	228	9	a	a	DET
cbi-57	228	10	boundless	boundless	ADJ
cbi-57	228	11	number	number	NOUN
cbi-57	228	12	.	.	PUNCT
cbi-57	229	1	that	that	PRON
cbi-57	229	2	is	be	AUX
cbi-57	229	3	,	,	PUNCT
cbi-57	229	4	there	there	PRON
cbi-57	229	5	is	be	VERB
cbi-57	229	6	not	not	PART
cbi-57	229	7	"	"	PUNCT
cbi-57	229	8	+1	+1	PROPN
cbi-57	229	9	that	that	PRON
cbi-57	229	10	is	be	AUX
cbi-57	229	11	distinct	distinct	ADJ
cbi-57	229	12	from	from	ADP
cbi-57	229	13	"	"	PUNCT
cbi-57	229	14	.	.	PUNCT
cbi-57	230	1	likewise	likewise	ADV
cbi-57	230	2	,	,	PUNCT
cbi-57	230	3	"	"	PUNCT
cbi-57	230	4	#	#	SYM
cbi-57	230	5	1	1	NUM
cbi-57	230	6	is	be	AUX
cbi-57	230	7	also	also	ADV
cbi-57	230	8	infinity	infinity	NOUN
cbi-57	230	9	.	.	PUNCT
cbi-57	231	1	essentially	essentially	ADV
cbi-57	231	2	,	,	PUNCT
cbi-57	231	3	within	within	ADP
cbi-57	231	4	the	the	DET
cbi-57	231	5	real	real	ADJ
cbi-57	231	6	numbers	number	NOUN
cbi-57	231	7	,	,	PUNCT
cbi-57	231	8	"	"	PUNCT
cbi-57	231	9	is	be	AUX
cbi-57	231	10	used	use	VERB
cbi-57	231	11	largely	largely	ADV
cbi-57	231	12	like	like	ADP
cbi-57	231	13	an	an	DET
cbi-57	231	14	ambiguous	ambiguous	ADJ
cbi-57	231	15	infinite	infinite	ADJ
cbi-57	231	16	value	value	NOUN
cbi-57	231	17	,	,	PUNCT
cbi-57	231	18	essentially	essentially	ADV
cbi-57	231	19	saying	say	VERB
cbi-57	231	20	that	that	SCONJ
cbi-57	231	21	“	"	PUNCT
cbi-57	231	22	the	the	DET
cbi-57	231	23	real	real	ADJ
cbi-57	231	24	numbers	number	NOUN
cbi-57	231	25	ca	can	AUX
cbi-57	231	26	n’t	not	PART
cbi-57	231	27	handle	handle	VERB
cbi-57	231	28	this	this	DET
cbi-57	231	29	value	value	NOUN
cbi-57	231	30	.	.	PUNCT
cbi-57	231	31	”	"	PUNCT
cbi-57	232	1	if	if	SCONJ
cbi-57	232	2	,	,	PUNCT
cbi-57	232	3	instead	instead	ADV
cbi-57	232	4	,	,	PUNCT
cbi-57	232	5	the	the	DET
cbi-57	232	6	hyperreal	hyperreal	ADJ
cbi-57	232	7	numbers	number	NOUN
cbi-57	232	8	are	be	AUX
cbi-57	232	9	used	use	VERB
cbi-57	232	10	,	,	PUNCT
cbi-57	232	11	then	then	ADV
cbi-57	232	12	!	!	PUNCT
cbi-57	233	1	and	and	CCONJ
cbi-57	233	2	!	!	PUNCT
cbi-57	234	1	+	+	CCONJ
cbi-57	234	2	1	1	NUM
cbi-57	234	3	are	be	AUX
cbi-57	234	4	distinct	distinct	ADJ
cbi-57	234	5	quantities	quantity	NOUN
cbi-57	234	6	,	,	PUNCT
cbi-57	234	7	despite	despite	SCONJ
cbi-57	234	8	the	the	DET
cbi-57	234	9	fact	fact	NOUN
cbi-57	234	10	that	that	SCONJ
cbi-57	234	11	they	they	PRON
cbi-57	234	12	are	be	AUX
cbi-57	234	13	both	both	PRON
cbi-57	234	14	infinite	infinite	ADJ
cbi-57	234	15	.	.	PUNCT
cbi-57	235	1	the	the	DET
cbi-57	235	2	rules	rule	NOUN
cbi-57	235	3	for	for	ADP
cbi-57	235	4	manipulating	manipulate	VERB
cbi-57	235	5	series	series	NOUN
cbi-57	235	6	come	come	VERB
cbi-57	235	7	from	from	ADP
cbi-57	235	8	these	these	DET
cbi-57	235	9	ideas	idea	NOUN
cbi-57	235	10	.	.	PUNCT
cbi-57	236	1	see	see	VERB
cbi-57	236	2	section	section	NOUN
cbi-57	236	3	11	11	NUM
cbi-57	236	4	for	for	ADP
cbi-57	236	5	a	a	DET
cbi-57	236	6	possible	possible	ADJ
cbi-57	236	7	exception	exception	NOUN
cbi-57	236	8	to	to	ADP
cbi-57	236	9	these	these	DET
cbi-57	236	10	rules	rule	NOUN
cbi-57	236	11	.	.	PUNCT
cbi-57	237	1	8.1	8.1	NUM
cbi-57	237	2	finite	finite	ADJ
cbi-57	237	3	term	term	NOUN
cbi-57	237	4	addition	addition	NOUN
cbi-57	237	5	to	to	PART
cbi-57	237	6	begin	begin	VERB
cbi-57	237	7	with	with	ADP
cbi-57	237	8	,	,	PUNCT
cbi-57	237	9	it	it	PRON
cbi-57	237	10	is	be	AUX
cbi-57	237	11	possible	possible	ADJ
cbi-57	237	12	to	to	PART
cbi-57	237	13	easily	easily	ADV
cbi-57	237	14	add	add	VERB
cbi-57	237	15	a	a	DET
cbi-57	237	16	scalar	scalar	ADJ
cbi-57	237	17	value	value	NOUN
cbi-57	237	18	to	to	ADP
cbi-57	237	19	a	a	DET
cbi-57	237	20	series	series	NOUN
cbi-57	237	21	,	,	PUNCT
cbi-57	237	22	provided	provide	VERB
cbi-57	237	23	that	that	SCONJ
cbi-57	237	24	it	it	PRON
cbi-57	237	25	is	be	AUX
cbi-57	237	26	added	add	VERB
cbi-57	237	27	to	to	ADP
cbi-57	237	28	one	one	NUM
cbi-57	237	29	of	of	ADP
cbi-57	237	30	the	the	DET
cbi-57	237	31	particular	particular	ADJ
cbi-57	237	32	terms	term	NOUN
cbi-57	237	33	of	of	ADP
cbi-57	237	34	the	the	DET
cbi-57	237	35	series	series	NOUN
cbi-57	237	36	.	.	PUNCT
cbi-57	238	1	in	in	ADP
cbi-57	238	2	other	other	ADJ
cbi-57	238	3	words	word	NOUN
cbi-57	238	4	,	,	PUNCT
cbi-57	238	5	suppose	suppose	VERB
cbi-57	238	6	the	the	DET
cbi-57	238	7	value	value	NOUN
cbi-57	238	8	a	a	PRON
cbi-57	238	9	is	be	AUX
cbi-57	238	10	added	add	VERB
cbi-57	238	11	to	to	ADP
cbi-57	238	12	the	the	DET
cbi-57	238	13	series	series	NOUN
cbi-57	238	14	1	1	NUM
cbi-57	238	15	+	+	SYM
cbi-57	238	16	2	2	NUM
cbi-57	238	17	+	+	NUM
cbi-57	238	18	3	3	NUM
cbi-57	238	19	+	+	NUM
cbi-57	238	20	.	.	PUNCT
cbi-57	238	21	.	.	PUNCT
cbi-57	239	1	..	..	PUNCT
cbi-57	239	2	this	this	PRON
cbi-57	239	3	can	can	AUX
cbi-57	239	4	be	be	AUX
cbi-57	239	5	written	write	VERB
cbi-57	239	6	as	as	ADP
cbi-57	239	7	a	a	DET
cbi-57	239	8	+	+	NOUN
cbi-57	239	9	!	!	PUNCT
cbi-57	239	10	!	!	PUNCT
cbi-57	240	1	i=1	i=1	PROPN
cbi-57	241	1	i	i	PRON
cbi-57	241	2	(	(	PUNCT
cbi-57	241	3	23	23	NUM
cbi-57	241	4	)	)	PUNCT
cbi-57	241	5	or	or	CCONJ
cbi-57	241	6	as	as	ADP
cbi-57	241	7	a	a	DET
cbi-57	241	8	+	+	X
cbi-57	241	9	(	(	PUNCT
cbi-57	241	10	1	1	NUM
cbi-57	241	11	+	+	NUM
cbi-57	241	12	2	2	NUM
cbi-57	241	13	+	+	NUM
cbi-57	241	14	3	3	NUM
cbi-57	241	15	+	+	NUM
cbi-57	241	16	.	.	PUNCT
cbi-57	241	17	.	.	PUNCT
cbi-57	241	18	.	.	PUNCT
cbi-57	241	19	)	)	PUNCT
cbi-57	241	20	.	.	PUNCT
cbi-57	242	1	(	(	PUNCT
cbi-57	242	2	24	24	NUM
cbi-57	242	3	)	)	PUNCT
cbi-57	242	4	to	to	PART
cbi-57	242	5	integrate	integrate	VERB
cbi-57	242	6	a	a	PRON
cbi-57	242	7	into	into	ADP
cbi-57	242	8	the	the	DET
cbi-57	242	9	series	series	NOUN
cbi-57	242	10	,	,	PUNCT
cbi-57	242	11	a	a	PRON
cbi-57	242	12	can	can	AUX
cbi-57	242	13	be	be	AUX
cbi-57	242	14	added	add	VERB
cbi-57	242	15	to	to	ADP
cbi-57	242	16	any	any	DET
cbi-57	242	17	distinct	distinct	ADJ
cbi-57	242	18	position	position	NOUN
cbi-57	242	19	.	.	PUNCT
cbi-57	243	1	the	the	DET
cbi-57	243	2	series	series	NOUN
cbi-57	243	3	could	could	AUX
cbi-57	243	4	read	read	VERB
cbi-57	243	5	as	as	ADP
cbi-57	243	6	(	(	PUNCT
cbi-57	243	7	a	a	DET
cbi-57	243	8	+	+	NOUN
cbi-57	243	9	1	1	NUM
cbi-57	243	10	)	)	PUNCT
cbi-57	243	11	+	+	CCONJ
cbi-57	243	12	2	2	NUM
cbi-57	243	13	+	+	NUM
cbi-57	243	14	3	3	NUM
cbi-57	243	15	+	+	NUM
cbi-57	243	16	.	.	PUNCT
cbi-57	243	17	.	.	PUNCT
cbi-57	243	18	.	.	PUNCT
cbi-57	244	1	(	(	PUNCT
cbi-57	244	2	25	25	NUM
cbi-57	244	3	)	)	PUNCT
cbi-57	244	4	or	or	CCONJ
cbi-57	244	5	1	1	NUM
cbi-57	244	6	+	+	SYM
cbi-57	244	7	2	2	NUM
cbi-57	244	8	+	+	CCONJ
cbi-57	244	9	(	(	PUNCT
cbi-57	244	10	a	a	PRON
cbi-57	244	11	+	+	NOUN
cbi-57	244	12	3	3	NUM
cbi-57	244	13	)	)	PUNCT
cbi-57	244	14	+	+	PUNCT
cbi-57	244	15	.	.	PUNCT
cbi-57	244	16	.	.	PUNCT
cbi-57	244	17	.	.	PUNCT
cbi-57	244	18	.	.	PUNCT
cbi-57	245	1	(	(	PUNCT
cbi-57	245	2	26	26	NUM
cbi-57	245	3	)	)	PUNCT
cbi-57	245	4	all	all	PRON
cbi-57	245	5	of	of	ADP
cbi-57	245	6	these	these	DET
cbi-57	245	7	yield	yield	NOUN
cbi-57	245	8	the	the	DET
cbi-57	245	9	same	same	ADJ
cbi-57	245	10	value	value	NOUN
cbi-57	245	11	for	for	ADP
cbi-57	245	12	the	the	DET
cbi-57	245	13	final	final	ADJ
cbi-57	245	14	series	series	NOUN
cbi-57	245	15	,	,	PUNCT
cbi-57	245	16	as	as	ADV
cbi-57	245	17	long	long	ADV
cbi-57	245	18	as	as	SCONJ
cbi-57	245	19	partial	partial	ADJ
cbi-57	245	20	sums	sum	NOUN
cbi-57	245	21	are	be	AUX
cbi-57	245	22	taken	take	VERB
cbi-57	245	23	starting	start	VERB
cbi-57	245	24	after	after	ADP
cbi-57	245	25	the	the	DET
cbi-57	245	26	index	index	NOUN
cbi-57	245	27	where	where	SCONJ
cbi-57	245	28	a	a	PRON
cbi-57	245	29	is	be	AUX
cbi-57	245	30	added	add	VERB
cbi-57	245	31	.	.	PUNCT
cbi-57	246	1	additionally	additionally	ADV
cbi-57	246	2	,	,	PUNCT
cbi-57	246	3	a	a	PRON
cbi-57	246	4	can	can	AUX
cbi-57	246	5	be	be	AUX
cbi-57	246	6	spread	spread	VERB
cbi-57	246	7	across	across	ADP
cbi-57	246	8	multiple	multiple	ADJ
cbi-57	246	9	finite	finite	ADJ
cbi-57	246	10	terms	term	NOUN
cbi-57	246	11	.	.	PUNCT
cbi-57	247	1	for	for	ADP
cbi-57	247	2	instance	instance	NOUN
cbi-57	247	3	,	,	PUNCT
cbi-57	247	4	half	half	NOUN
cbi-57	247	5	of	of	ADP
cbi-57	247	6	a	a	PRON
cbi-57	247	7	could	could	AUX
cbi-57	247	8	be	be	AUX
cbi-57	247	9	added	add	VERB
cbi-57	247	10	to	to	ADP
cbi-57	247	11	each	each	PRON
cbi-57	247	12	of	of	ADP
cbi-57	247	13	the	the	DET
cbi-57	247	14	first	first	ADJ
cbi-57	247	15	two	two	NUM
cbi-57	247	16	terms	term	NOUN
cbi-57	247	17	,	,	PUNCT
cbi-57	247	18	yielding	yield	VERB
cbi-57	247	19	a	a	DET
cbi-57	247	20	+	+	X
cbi-57	247	21	(	(	PUNCT
cbi-57	247	22	1	1	NUM
cbi-57	247	23	+	+	NUM
cbi-57	247	24	2	2	NUM
cbi-57	247	25	+	+	NUM
cbi-57	247	26	3	3	NUM
cbi-57	247	27	+	+	NUM
cbi-57	247	28	.	.	PUNCT
cbi-57	247	29	.	.	PUNCT
cbi-57	247	30	.	.	PUNCT
cbi-57	247	31	)	)	PUNCT
cbi-57	248	1	=	=	PRON
cbi-57	248	2	'	'	PUNCT
cbi-57	248	3	1	1	NUM
cbi-57	248	4	+	+	CCONJ
cbi-57	248	5	a	a	DET
cbi-57	248	6	2	2	NUM
cbi-57	248	7	(	(	PUNCT
cbi-57	248	8	+	+	NOUN
cbi-57	248	9	'	'	PUNCT
cbi-57	248	10	2	2	NUM
cbi-57	248	11	+	+	CCONJ
cbi-57	248	12	a	a	DET
cbi-57	248	13	2	2	NUM
cbi-57	248	14	(	(	PUNCT
cbi-57	248	15	+	+	NUM
cbi-57	248	16	3	3	NUM
cbi-57	248	17	+	+	NUM
cbi-57	248	18	.	.	PUNCT
cbi-57	248	19	.	.	PUNCT
cbi-57	248	20	.	.	PUNCT
cbi-57	248	21	.	.	PUNCT
cbi-57	249	1	(	(	PUNCT
cbi-57	249	2	27	27	NUM
cbi-57	249	3	)	)	PUNCT
cbi-57	249	4	in	in	ADP
cbi-57	249	5	fact	fact	NOUN
cbi-57	249	6	,	,	PUNCT
cbi-57	249	7	there	there	PRON
cbi-57	249	8	is	be	VERB
cbi-57	249	9	no	no	DET
cbi-57	249	10	reason	reason	NOUN
cbi-57	249	11	why	why	SCONJ
cbi-57	249	12	the	the	DET
cbi-57	249	13	same	same	ADJ
cbi-57	249	14	amount	amount	NOUN
cbi-57	249	15	would	would	AUX
cbi-57	249	16	have	have	VERB
cbi-57	249	17	to	to	PART
cbi-57	249	18	be	be	AUX
cbi-57	249	19	distributed	distribute	VERB
cbi-57	249	20	to	to	ADP
cbi-57	249	21	each	each	DET
cbi-57	249	22	position	position	NOUN
cbi-57	249	23	.	.	PUNCT
cbi-57	250	1	a	a	DET
cbi-57	250	2	+	+	X
cbi-57	250	3	(	(	PUNCT
cbi-57	250	4	1	1	NUM
cbi-57	250	5	+	+	NUM
cbi-57	250	6	2	2	NUM
cbi-57	250	7	+	+	NUM
cbi-57	250	8	3	3	NUM
cbi-57	250	9	+	+	NUM
cbi-57	250	10	.	.	PUNCT
cbi-57	250	11	.	.	PUNCT
cbi-57	250	12	.	.	PUNCT
cbi-57	250	13	)	)	PUNCT
cbi-57	251	1	=	=	PUNCT
cbi-57	251	2	#	#	NOUN
cbi-57	251	3	1	1	NUM
cbi-57	251	4	+	+	CCONJ
cbi-57	251	5	2	2	NUM
cbi-57	251	6	5	5	NUM
cbi-57	251	7	a	a	DET
cbi-57	251	8	$	$	SYM
cbi-57	251	9	+	+	NUM
cbi-57	251	10	#	#	SYM
cbi-57	251	11	2	2	NUM
cbi-57	251	12	+	+	CCONJ
cbi-57	251	13	3	3	NUM
cbi-57	251	14	5	5	NUM
cbi-57	251	15	a	a	DET
cbi-57	251	16	$	$	SYM
cbi-57	251	17	+	+	NUM
cbi-57	251	18	3	3	NUM
cbi-57	251	19	+	+	NUM
cbi-57	251	20	.	.	PUNCT
cbi-57	251	21	.	.	PUNCT
cbi-57	251	22	.	.	PUNCT
cbi-57	252	1	(	(	PUNCT
cbi-57	252	2	28	28	NUM
cbi-57	252	3	)	)	PUNCT
cbi-57	252	4	8.2	8.2	NUM
cbi-57	252	5	finite	finite	ADJ
cbi-57	252	6	term	term	NOUN
cbi-57	252	7	insertion	insertion	NOUN
cbi-57	252	8	and	and	CCONJ
cbi-57	252	9	removal	removal	NOUN
cbi-57	252	10	because	because	SCONJ
cbi-57	252	11	this	this	DET
cbi-57	252	12	method	method	NOUN
cbi-57	252	13	of	of	ADP
cbi-57	252	14	summation	summation	NOUN
cbi-57	252	15	is	be	AUX
cbi-57	252	16	based	base	VERB
cbi-57	252	17	on	on	ADP
cbi-57	252	18	partial	partial	ADJ
cbi-57	252	19	sums	sum	NOUN
cbi-57	252	20	,	,	PUNCT
cbi-57	252	21	it	it	PRON
cbi-57	252	22	should	should	AUX
cbi-57	252	23	be	be	AUX
cbi-57	252	24	apparent	apparent	ADJ
cbi-57	252	25	that	that	SCONJ
cbi-57	252	26	inserting	insert	VERB
cbi-57	252	27	and	and	CCONJ
cbi-57	252	28	removing	remove	VERB
cbi-57	252	29	terms	term	NOUN
cbi-57	252	30	will	will	AUX
cbi-57	252	31	in	in	ADP
cbi-57	252	32	fact	fact	NOUN
cbi-57	252	33	alter	alter	VERB
cbi-57	252	34	the	the	DET
cbi-57	252	35	summation	summation	NOUN
cbi-57	252	36	.	.	PUNCT
cbi-57	253	1	for	for	ADP
cbi-57	253	2	instance	instance	NOUN
cbi-57	253	3	,	,	PUNCT
cbi-57	253	4	let	let	VERB
cbi-57	253	5	’s	’s	PRON
cbi-57	253	6	begin	begin	VERB
cbi-57	253	7	with	with	ADP
cbi-57	253	8	the	the	DET
cbi-57	253	9	arithmetic	arithmetic	ADJ
cbi-57	253	10	sum	sum	NOUN
cbi-57	253	11	1	1	NUM
cbi-57	254	1	+	+	NOUN
cbi-57	254	2	1	1	NUM
cbi-57	254	3	+	+	SYM
cbi-57	254	4	1	1	NUM
cbi-57	254	5	+	+	NUM
cbi-57	254	6	.	.	PUNCT
cbi-57	254	7	.	.	PUNCT
cbi-57	255	1	..	..	PUNCT
cbi-57	256	1	it	it	PRON
cbi-57	256	2	may	may	AUX
cbi-57	256	3	seem	seem	VERB
cbi-57	256	4	intuitive	intuitive	ADJ
cbi-57	256	5	that	that	SCONJ
cbi-57	256	6	one	one	PRON
cbi-57	256	7	should	should	AUX
cbi-57	256	8	be	be	AUX
cbi-57	256	9	able	able	ADJ
cbi-57	256	10	to	to	PART
cbi-57	256	11	freely	freely	ADV
cbi-57	256	12	add	add	VERB
cbi-57	256	13	or	or	CCONJ
cbi-57	256	14	remove	remove	VERB
cbi-57	256	15	a	a	DET
cbi-57	256	16	1	1	NUM
cbi-57	256	17	from	from	ADP
cbi-57	256	18	this	this	DET
cbi-57	256	19	sum	sum	NOUN
cbi-57	256	20	without	without	ADP
cbi-57	256	21	a!ecting	a!ecte	VERB
cbi-57	256	22	the	the	DET
cbi-57	256	23	sum	sum	NOUN
cbi-57	256	24	.	.	PUNCT
cbi-57	257	1	in	in	ADP
cbi-57	257	2	this	this	DET
cbi-57	257	3	particular	particular	ADJ
cbi-57	257	4	series	series	NOUN
cbi-57	257	5	,	,	PUNCT
cbi-57	257	6	the	the	DET
cbi-57	257	7	exact	exact	ADJ
cbi-57	257	8	hyperreal	hyperreal	NOUN
cbi-57	257	9	value	value	NOUN
cbi-57	257	10	does	do	AUX
cbi-57	257	11	change	change	VERB
cbi-57	257	12	,	,	PUNCT
cbi-57	257	13	but	but	CCONJ
cbi-57	257	14	not	not	PART
cbi-57	257	15	the	the	DET
cbi-57	257	16	principal	principal	ADJ
cbi-57	257	17	value	value	NOUN
cbi-57	257	18	.	.	PUNCT
cbi-57	258	1	again	again	ADV
cbi-57	258	2	,	,	PUNCT
cbi-57	258	3	remember	remember	VERB
cbi-57	258	4	that	that	SCONJ
cbi-57	258	5	,	,	PUNCT
cbi-57	258	6	as	as	SCONJ
cbi-57	258	7	mentioned	mention	VERB
cbi-57	258	8	in	in	ADP
cbi-57	258	9	section	section	NOUN
cbi-57	258	10	3	3	NUM
cbi-57	258	11	,	,	PUNCT
cbi-57	258	12	this	this	DET
cbi-57	258	13	conception	conception	NOUN
cbi-57	258	14	of	of	ADP
cbi-57	258	15	summation	summation	NOUN
cbi-57	258	16	will	will	AUX
cbi-57	258	17	be	be	AUX
cbi-57	258	18	based	base	VERB
cbi-57	258	19	on	on	ADP
cbi-57	258	20	partial	partial	ADJ
cbi-57	258	21	sums	sum	NOUN
cbi-57	258	22	.	.	PUNCT
cbi-57	259	1	so	so	ADV
cbi-57	259	2	,	,	PUNCT
cbi-57	259	3	let	let	VERB
cbi-57	259	4	us	we	PRON
cbi-57	259	5	begin	begin	VERB
cbi-57	259	6	by	by	ADP
cbi-57	259	7	considering	consider	VERB
cbi-57	259	8	the	the	DET
cbi-57	259	9	partial	partial	ADJ
cbi-57	259	10	sum	sum	NOUN
cbi-57	259	11	k	k	NOUN
cbi-57	259	12	!	!	PUNCT
cbi-57	260	1	i=1	i=1	PROPN
cbi-57	261	1	1	1	NUM
cbi-57	261	2	.	.	PUNCT
cbi-57	261	3	(	(	PUNCT
cbi-57	261	4	29	29	NUM
cbi-57	261	5	)	)	PUNCT
cbi-57	261	6	if	if	SCONJ
cbi-57	261	7	k	k	PROPN
cbi-57	261	8	is	be	AUX
cbi-57	261	9	a	a	DET
cbi-57	261	10	finite	finite	ADJ
cbi-57	261	11	number	number	NOUN
cbi-57	261	12	,	,	PUNCT
cbi-57	261	13	then	then	ADV
cbi-57	261	14	adding	add	VERB
cbi-57	261	15	one	one	NUM
cbi-57	261	16	to	to	ADP
cbi-57	261	17	this	this	DET
cbi-57	261	18	sequence	sequence	NOUN
cbi-57	261	19	will	will	AUX
cbi-57	261	20	in	in	ADP
cbi-57	261	21	fact	fact	NOUN
cbi-57	261	22	alter	alter	VERB
cbi-57	261	23	its	its	PRON
cbi-57	261	24	value	value	NOUN
cbi-57	261	25	.	.	PUNCT
cbi-57	262	1	additionally	additionally	ADV
cbi-57	262	2	,	,	PUNCT
cbi-57	262	3	removing	remove	VERB
cbi-57	262	4	a	a	DET
cbi-57	262	5	1	1	NUM
cbi-57	262	6	from	from	ADP
cbi-57	262	7	this	this	DET
cbi-57	262	8	sequence	sequence	NOUN
cbi-57	262	9	will	will	AUX
cbi-57	262	10	also	also	ADV
cbi-57	262	11	alter	alter	VERB
cbi-57	262	12	its	its	PRON
cbi-57	262	13	value	value	NOUN
cbi-57	262	14	.	.	PUNCT
cbi-57	263	1	therefore	therefore	ADV
cbi-57	263	2	,	,	PUNCT
cbi-57	263	3	k	k	X
cbi-57	263	4	!	!	PUNCT
cbi-57	263	5	i=1	i=1	PROPN
cbi-57	263	6	1	1	NUM
cbi-57	263	7	!	!	NUM
cbi-57	263	8	1	1	NUM
cbi-57	264	1	+	+	CCONJ
cbi-57	264	2	k	k	X
cbi-57	264	3	!	!	PUNCT
cbi-57	264	4	i=1	i=1	PROPN
cbi-57	265	1	1	1	NUM
cbi-57	265	2	.	.	PUNCT
cbi-57	265	3	(	(	PUNCT
cbi-57	265	4	30	30	NUM
cbi-57	265	5	)	)	PUNCT
cbi-57	265	6	likewise	likewise	ADV
cbi-57	265	7	,	,	PUNCT
cbi-57	265	8	k	k	X
cbi-57	265	9	!	!	PUNCT
cbi-57	266	1	i=1	i=1	PROPN
cbi-57	266	2	1	1	NUM
cbi-57	266	3	!	!	PUNCT
cbi-57	267	1	k	k	X
cbi-57	267	2	!	!	PUNCT
cbi-57	268	1	i=0	i=0	PROPN
cbi-57	268	2	1	1	NUM
cbi-57	268	3	!	!	PUNCT
cbi-57	269	1	k	k	X
cbi-57	269	2	!	!	PUNCT
cbi-57	269	3	i=2	i=2	PROPN
cbi-57	269	4	1	1	NUM
cbi-57	269	5	.	.	PUNCT
cbi-57	270	1	(	(	PUNCT
cbi-57	270	2	31	31	NUM
cbi-57	270	3	)	)	PUNCT
cbi-57	270	4	because	because	SCONJ
cbi-57	270	5	performing	perform	VERB
cbi-57	270	6	these	these	DET
cbi-57	270	7	operations	operation	NOUN
cbi-57	270	8	will	will	AUX
cbi-57	270	9	change	change	VERB
cbi-57	270	10	the	the	DET
cbi-57	270	11	value	value	NOUN
cbi-57	270	12	for	for	ADP
cbi-57	270	13	any	any	DET
cbi-57	270	14	partial	partial	ADJ
cbi-57	270	15	sum	sum	NOUN
cbi-57	270	16	of	of	ADP
cbi-57	270	17	k	k	PROPN
cbi-57	270	18	terms	term	NOUN
cbi-57	270	19	for	for	ADP
cbi-57	270	20	a	a	DET
cbi-57	270	21	finite	finite	NOUN
cbi-57	270	22	k	k	NOUN
cbi-57	270	23	,	,	PUNCT
cbi-57	270	24	they	they	PRON
cbi-57	270	25	will	will	AUX
cbi-57	270	26	also	also	ADV
cbi-57	270	27	change	change	VERB
cbi-57	270	28	the	the	DET
cbi-57	270	29	value	value	NOUN
cbi-57	270	30	for	for	ADP
cbi-57	270	31	a	a	DET
cbi-57	270	32	hyperreal	hyperreal	NOUN
cbi-57	270	33	k	k	NOUN
cbi-57	270	34	such	such	ADJ
cbi-57	270	35	as	as	ADP
cbi-57	270	36	!	!	PUNCT
cbi-57	270	37	.	.	PUNCT
cbi-57	271	1	however	however	ADV
cbi-57	271	2	,	,	PUNCT
cbi-57	271	3	for	for	ADP
cbi-57	271	4	these	these	DET
cbi-57	271	5	particular	particular	ADJ
cbi-57	271	6	series	series	NOUN
cbi-57	271	7	,	,	PUNCT
cbi-57	271	8	the	the	DET
cbi-57	271	9	principal	principal	ADJ
cbi-57	271	10	value	value	NOUN
cbi-57	271	11	will	will	AUX
cbi-57	271	12	be	be	AUX
cbi-57	271	13	the	the	DET
cbi-57	271	14	same	same	ADJ
cbi-57	271	15	,	,	PUNCT
cbi-57	271	16	because	because	SCONJ
cbi-57	271	17	!	!	PUNCT
cbi-57	272	1	$	$	X
cbi-57	272	2	!	!	PUNCT
cbi-57	273	1	+	+	CCONJ
cbi-57	273	2	1	1	NUM
cbi-57	273	3	$	$	NUM
cbi-57	273	4	!	!	PUNCT
cbi-57	274	1	#	#	NOUN
cbi-57	274	2	1	1	NUM
cbi-57	274	3	.	.	PUNCT
cbi-57	275	1	additionally	additionally	ADV
cbi-57	275	2	,	,	PUNCT
cbi-57	275	3	a	a	DET
cbi-57	275	4	more	more	ADV
cbi-57	275	5	surpising	surpising	ADJ
cbi-57	275	6	fact	fact	NOUN
cbi-57	275	7	is	be	AUX
cbi-57	275	8	that	that	SCONJ
cbi-57	275	9	removing	remove	VERB
cbi-57	275	10	a	a	DET
cbi-57	275	11	term	term	NOUN
cbi-57	275	12	from	from	ADP
cbi-57	275	13	a	a	DET
cbi-57	275	14	sequence	sequence	NOUN
cbi-57	275	15	also	also	ADV
cbi-57	275	16	changes	change	VERB
cbi-57	275	17	its	its	PRON
cbi-57	275	18	value	value	NOUN
cbi-57	275	19	if	if	SCONJ
cbi-57	275	20	it	it	PRON
cbi-57	275	21	does	do	AUX
cbi-57	275	22	not	not	PART
cbi-57	275	23	also	also	ADV
cbi-57	275	24	change	change	VERB
cbi-57	275	25	the	the	DET
cbi-57	275	26	number	number	NOUN
cbi-57	275	27	of	of	ADP
cbi-57	275	28	terms	term	NOUN
cbi-57	275	29	being	be	AUX
cbi-57	275	30	summed	sum	VERB
cbi-57	275	31	.	.	PUNCT
cbi-57	276	1	consider	consider	VERB
cbi-57	276	2	the	the	DET
cbi-57	276	3	series	series	NOUN
cbi-57	276	4	1	1	NUM
cbi-57	276	5	+	+	SYM
cbi-57	276	6	2	2	NUM
cbi-57	276	7	+	+	NUM
cbi-57	276	8	3	3	NUM
cbi-57	276	9	+	+	NUM
cbi-57	276	10	.	.	PUNCT
cbi-57	276	11	.	.	PUNCT
cbi-57	276	12	.	.	PUNCT
cbi-57	277	1	=	=	PUNCT
cbi-57	277	2	!	!	PUNCT
cbi-57	277	3	!	!	PUNCT
cbi-57	278	1	i=1	i=1	PROPN
cbi-57	278	2	i.	i.	PROPN
cbi-57	278	3	(	(	PUNCT
cbi-57	278	4	32	32	NUM
cbi-57	278	5	)	)	PUNCT
cbi-57	278	6	this	this	DET
cbi-57	278	7	series	series	NOUN
cbi-57	278	8	is	be	AUX
cbi-57	278	9	not	not	PART
cbi-57	278	10	equal	equal	ADJ
cbi-57	278	11	to	to	ADP
cbi-57	278	12	the	the	DET
cbi-57	278	13	series	series	NOUN
cbi-57	278	14	1	1	NUM
cbi-57	278	15	+	+	CCONJ
cbi-57	278	16	!	!	PUNCT
cbi-57	278	17	!	!	PUNCT
cbi-57	279	1	i=1	i=1	PROPN
cbi-57	280	1	(	(	PUNCT
cbi-57	280	2	i	i	PRON
cbi-57	280	3	+	+	NOUN
cbi-57	280	4	1	1	NUM
cbi-57	280	5	)	)	PUNCT
cbi-57	280	6	.	.	PUNCT
cbi-57	281	1	(	(	PUNCT
cbi-57	281	2	33	33	NUM
cbi-57	281	3	)	)	SYM
cbi-57	281	4	12	12	NUM
cbi-57	281	5	hyperreal	hyperreal	ADJ
cbi-57	281	6	numbers	number	NOUN
cbi-57	281	7	for	for	ADP
cbi-57	281	8	infinite	infinite	ADJ
cbi-57	281	9	divergent	divergent	ADJ
cbi-57	281	10	series	series	NOUN
cbi-57	281	11	although	although	SCONJ
cbi-57	281	12	it	it	PRON
cbi-57	281	13	does	do	AUX
cbi-57	281	14	have	have	VERB
cbi-57	281	15	the	the	DET
cbi-57	281	16	same	same	ADJ
cbi-57	281	17	principal	principal	ADJ
cbi-57	281	18	value	value	NOUN
cbi-57	281	19	in	in	ADP
cbi-57	281	20	this	this	DET
cbi-57	281	21	case	case	NOUN
cbi-57	281	22	.	.	PUNCT
cbi-57	282	1	in	in	ADP
cbi-57	282	2	other	other	ADJ
cbi-57	282	3	words	word	NOUN
cbi-57	282	4	,	,	PUNCT
cbi-57	282	5	(	(	PUNCT
cbi-57	282	6	1	1	NUM
cbi-57	282	7	+	+	SYM
cbi-57	282	8	2	2	NUM
cbi-57	282	9	+	+	NUM
cbi-57	282	10	3	3	NUM
cbi-57	282	11	+	+	NUM
cbi-57	282	12	.	.	PUNCT
cbi-57	282	13	.	.	PUNCT
cbi-57	282	14	.	.	PUNCT
cbi-57	282	15	)	)	PUNCT
cbi-57	282	16	!	!	PUNCT
cbi-57	283	1	1	1	NUM
cbi-57	284	1	+	+	CCONJ
cbi-57	284	2	(	(	PUNCT
cbi-57	284	3	2	2	NUM
cbi-57	284	4	+	+	SYM
cbi-57	284	5	3	3	NUM
cbi-57	284	6	+	+	SYM
cbi-57	284	7	4	4	NUM
cbi-57	284	8	+	+	CCONJ
cbi-57	284	9	.	.	PUNCT
cbi-57	284	10	.	.	PUNCT
cbi-57	284	11	.	.	PUNCT
cbi-57	284	12	)	)	PUNCT
cbi-57	285	1	(	(	PUNCT
cbi-57	285	2	34	34	NUM
cbi-57	285	3	)	)	PUNCT
cbi-57	285	4	but	but	CCONJ
cbi-57	285	5	(	(	PUNCT
cbi-57	285	6	1	1	NUM
cbi-57	285	7	+	+	NUM
cbi-57	285	8	2	2	NUM
cbi-57	285	9	+	+	NUM
cbi-57	285	10	3	3	NUM
cbi-57	285	11	+	+	NUM
cbi-57	285	12	.	.	PUNCT
cbi-57	285	13	.	.	PUNCT
cbi-57	285	14	.	.	PUNCT
cbi-57	285	15	)	)	PUNCT
cbi-57	286	1	$	$	SYM
cbi-57	286	2	1	1	NUM
cbi-57	286	3	+	+	CCONJ
cbi-57	286	4	(	(	PUNCT
cbi-57	286	5	2	2	NUM
cbi-57	286	6	+	+	SYM
cbi-57	286	7	3	3	NUM
cbi-57	286	8	+	+	SYM
cbi-57	286	9	4	4	NUM
cbi-57	286	10	+	+	CCONJ
cbi-57	286	11	.	.	PUNCT
cbi-57	286	12	.	.	PUNCT
cbi-57	286	13	.	.	PUNCT
cbi-57	286	14	)	)	PUNCT
cbi-57	286	15	.	.	PUNCT
cbi-57	287	1	(	(	PUNCT
cbi-57	287	2	35	35	NUM
cbi-57	287	3	)	)	PUNCT
cbi-57	287	4	the	the	DET
cbi-57	287	5	reason	reason	NOUN
cbi-57	287	6	for	for	ADP
cbi-57	287	7	this	this	PRON
cbi-57	287	8	is	be	AUX
cbi-57	287	9	readily	readily	ADV
cbi-57	287	10	apparent	apparent	ADJ
cbi-57	287	11	when	when	SCONJ
cbi-57	287	12	considering	consider	VERB
cbi-57	287	13	how	how	SCONJ
cbi-57	287	14	these	these	DET
cbi-57	287	15	work	work	NOUN
cbi-57	287	16	in	in	ADP
cbi-57	287	17	terms	term	NOUN
cbi-57	287	18	of	of	ADP
cbi-57	287	19	partial	partial	ADJ
cbi-57	287	20	sums	sum	NOUN
cbi-57	287	21	.	.	PUNCT
cbi-57	288	1	if	if	SCONJ
cbi-57	288	2	the	the	DET
cbi-57	288	3	parameter	parameter	NOUN
cbi-57	288	4	k	k	PROPN
cbi-57	288	5	was	be	AUX
cbi-57	288	6	used	use	VERB
cbi-57	288	7	instead	instead	ADV
cbi-57	288	8	of	of	ADP
cbi-57	288	9	!	!	PUNCT
cbi-57	288	10	,	,	PUNCT
cbi-57	288	11	then	then	ADV
cbi-57	288	12	it	it	PRON
cbi-57	288	13	is	be	AUX
cbi-57	288	14	apparent	apparent	ADJ
cbi-57	288	15	that	that	SCONJ
cbi-57	288	16	the	the	DET
cbi-57	288	17	value	value	NOUN
cbi-57	288	18	of	of	ADP
cbi-57	288	19	(	(	PUNCT
cbi-57	288	20	33	33	NUM
cbi-57	288	21	)	)	PUNCT
cbi-57	288	22	actually	actually	ADV
cbi-57	288	23	has	have	VERB
cbi-57	288	24	an	an	DET
cbi-57	288	25	extra	extra	ADJ
cbi-57	288	26	term	term	NOUN
cbi-57	288	27	compared	compare	VERB
cbi-57	288	28	to	to	ADP
cbi-57	288	29	(	(	PUNCT
cbi-57	288	30	32	32	NUM
cbi-57	288	31	)	)	PUNCT
cbi-57	288	32	.	.	PUNCT
cbi-57	289	1	that	that	PRON
cbi-57	289	2	is	be	AUX
cbi-57	289	3	,	,	PUNCT
cbi-57	289	4	it	it	PRON
cbi-57	289	5	is	be	AUX
cbi-57	289	6	obvious	obvious	ADJ
cbi-57	289	7	that	that	SCONJ
cbi-57	289	8	5	5	X
cbi-57	289	9	!	!	PUNCT
cbi-57	289	10	i=1	i=1	X
cbi-57	290	1	i	i	PRON
cbi-57	290	2	!	!	PUNCT
cbi-57	291	1	1	1	NUM
cbi-57	292	1	+	+	NUM
cbi-57	292	2	5	5	X
cbi-57	292	3	!	!	PUNCT
cbi-57	292	4	i=1	i=1	PROPN
cbi-57	293	1	(	(	PUNCT
cbi-57	293	2	i	i	PRON
cbi-57	293	3	+	+	NOUN
cbi-57	293	4	1	1	NUM
cbi-57	293	5	)	)	PUNCT
cbi-57	293	6	.	.	PUNCT
cbi-57	294	1	(	(	PUNCT
cbi-57	294	2	36	36	NUM
cbi-57	294	3	)	)	PUNCT
cbi-57	294	4	this	this	PRON
cbi-57	294	5	can	can	AUX
cbi-57	294	6	also	also	ADV
cbi-57	294	7	be	be	AUX
cbi-57	294	8	seen	see	VERB
cbi-57	294	9	in	in	ADP
cbi-57	294	10	the	the	DET
cbi-57	294	11	results	result	NOUN
cbi-57	294	12	of	of	ADP
cbi-57	294	13	applying	apply	VERB
cbi-57	294	14	the	the	DET
cbi-57	294	15	arithmetic	arithmetic	ADJ
cbi-57	294	16	series	series	NOUN
cbi-57	294	17	formula	formula	NOUN
cbi-57	294	18	to	to	ADP
cbi-57	294	19	the	the	DET
cbi-57	294	20	two	two	NUM
cbi-57	294	21	series	series	NOUN
cbi-57	294	22	.	.	PUNCT
cbi-57	295	1	for	for	ADP
cbi-57	295	2	(	(	PUNCT
cbi-57	295	3	1	1	NUM
cbi-57	295	4	+	+	NUM
cbi-57	295	5	2	2	NUM
cbi-57	295	6	+	+	NUM
cbi-57	295	7	3	3	NUM
cbi-57	295	8	+	+	NUM
cbi-57	295	9	.	.	PUNCT
cbi-57	295	10	.	.	PUNCT
cbi-57	295	11	.	.	PUNCT
cbi-57	295	12	)	)	PUNCT
cbi-57	296	1	the	the	DET
cbi-57	296	2	formula	formula	NOUN
cbi-57	296	3	yields	yield	VERB
cbi-57	296	4	!	!	PUNCT
cbi-57	296	5	2	2	NUM
cbi-57	296	6	2	2	NUM
cbi-57	296	7	+	+	CCONJ
cbi-57	296	8	!	!	PUNCT
cbi-57	297	1	2	2	X
cbi-57	297	2	.	.	PUNCT
cbi-57	298	1	however	however	ADV
cbi-57	298	2	,	,	PUNCT
cbi-57	298	3	for	for	ADP
cbi-57	298	4	(	(	PUNCT
cbi-57	298	5	2	2	NUM
cbi-57	298	6	+	+	NUM
cbi-57	298	7	3	3	NUM
cbi-57	298	8	+	+	NUM
cbi-57	298	9	4	4	NUM
cbi-57	298	10	+	+	NUM
cbi-57	298	11	.	.	PUNCT
cbi-57	298	12	.	.	PUNCT
cbi-57	298	13	.	.	PUNCT
cbi-57	298	14	)	)	PUNCT
cbi-57	299	1	the	the	DET
cbi-57	299	2	formula	formula	NOUN
cbi-57	299	3	yields	yield	VERB
cbi-57	299	4	!	!	PUNCT
cbi-57	300	1	2	2	NUM
cbi-57	300	2	2	2	NUM
cbi-57	300	3	+	+	CCONJ
cbi-57	300	4	3	3	NUM
cbi-57	300	5	2	2	NUM
cbi-57	300	6	!	!	PUNCT
cbi-57	300	7	.	.	PUNCT
cbi-57	301	1	now	now	ADV
cbi-57	301	2	,	,	PUNCT
cbi-57	301	3	terms	term	NOUN
cbi-57	301	4	can	can	AUX
cbi-57	301	5	be	be	AUX
cbi-57	301	6	removed	remove	VERB
cbi-57	301	7	without	without	ADP
cbi-57	301	8	even	even	ADV
cbi-57	301	9	a!ecting	a!ecte	VERB
cbi-57	301	10	the	the	DET
cbi-57	301	11	exact	exact	ADJ
cbi-57	301	12	hyperreal	hyperreal	NOUN
cbi-57	301	13	value	value	NOUN
cbi-57	301	14	if	if	SCONJ
cbi-57	301	15	they	they	PRON
cbi-57	301	16	are	be	AUX
cbi-57	301	17	replaced	replace	VERB
cbi-57	301	18	by	by	ADP
cbi-57	301	19	zeroes	zero	NOUN
cbi-57	301	20	in	in	ADP
cbi-57	301	21	the	the	DET
cbi-57	301	22	sequence	sequence	NOUN
cbi-57	301	23	,	,	PUNCT
cbi-57	301	24	or	or	CCONJ
cbi-57	301	25	if	if	SCONJ
cbi-57	301	26	the	the	DET
cbi-57	301	27	sequence	sequence	NOUN
cbi-57	301	28	starting	start	VERB
cbi-57	301	29	index	index	NOUN
cbi-57	301	30	is	be	AUX
cbi-57	301	31	moved	move	VERB
cbi-57	301	32	appropriately	appropriately	ADV
cbi-57	301	33	.	.	PUNCT
cbi-57	302	1	in	in	ADP
cbi-57	302	2	other	other	ADJ
cbi-57	302	3	words	word	NOUN
cbi-57	302	4	,	,	PUNCT
cbi-57	302	5	(	(	PUNCT
cbi-57	302	6	1	1	NUM
cbi-57	302	7	+	+	SYM
cbi-57	302	8	2	2	NUM
cbi-57	302	9	+	+	NUM
cbi-57	302	10	3	3	NUM
cbi-57	302	11	+	+	NUM
cbi-57	302	12	.	.	PUNCT
cbi-57	302	13	.	.	PUNCT
cbi-57	302	14	.	.	PUNCT
cbi-57	302	15	)	)	PUNCT
cbi-57	303	1	=	=	PUNCT
cbi-57	303	2	1	1	NUM
cbi-57	303	3	+	+	CCONJ
cbi-57	303	4	(	(	PUNCT
cbi-57	303	5	0	0	NUM
cbi-57	303	6	+	+	NUM
cbi-57	303	7	2	2	NUM
cbi-57	303	8	+	+	NUM
cbi-57	303	9	3	3	NUM
cbi-57	303	10	+	+	NUM
cbi-57	303	11	.	.	PUNCT
cbi-57	303	12	.	.	PUNCT
cbi-57	303	13	.	.	PUNCT
cbi-57	303	14	)	)	PUNCT
cbi-57	304	1	=	=	PUNCT
cbi-57	304	2	1	1	NUM
cbi-57	304	3	+	+	CCONJ
cbi-57	304	4	!	!	PUNCT
cbi-57	304	5	!	!	PUNCT
cbi-57	305	1	i=2	i=2	PROPN
cbi-57	305	2	i.	i.	PROPN
cbi-57	305	3	(	(	PUNCT
cbi-57	305	4	37	37	NUM
cbi-57	305	5	)	)	PUNCT
cbi-57	305	6	this	this	PRON
cbi-57	305	7	can	can	AUX
cbi-57	305	8	be	be	AUX
cbi-57	305	9	easily	easily	ADV
cbi-57	305	10	proved	prove	VERB
cbi-57	305	11	using	use	VERB
cbi-57	305	12	the	the	DET
cbi-57	305	13	principle	principle	NOUN
cbi-57	305	14	derived	derive	VERB
cbi-57	305	15	in	in	ADP
cbi-57	305	16	section	section	NOUN
cbi-57	305	17	8.1	8.1	NUM
cbi-57	305	18	.	.	PUNCT
cbi-57	306	1	for	for	ADP
cbi-57	306	2	instance	instance	NOUN
cbi-57	306	3	,	,	PUNCT
cbi-57	306	4	to	to	PART
cbi-57	306	5	move	move	VERB
cbi-57	306	6	the	the	DET
cbi-57	306	7	1	1	NUM
cbi-57	306	8	outside	outside	ADP
cbi-57	306	9	of	of	ADP
cbi-57	306	10	the	the	DET
cbi-57	306	11	series	series	NOUN
cbi-57	306	12	,	,	PUNCT
cbi-57	306	13	1	1	NUM
cbi-57	306	14	+	+	CCONJ
cbi-57	306	15	#	#	SYM
cbi-57	306	16	1	1	NUM
cbi-57	306	17	can	can	AUX
cbi-57	306	18	be	be	AUX
cbi-57	306	19	added	add	VERB
cbi-57	306	20	to	to	ADP
cbi-57	306	21	the	the	DET
cbi-57	306	22	series	series	NOUN
cbi-57	306	23	.	.	PUNCT
cbi-57	307	1	1	1	NUM
cbi-57	308	1	+	+	CCONJ
cbi-57	308	2	#	#	SYM
cbi-57	308	3	1	1	NUM
cbi-57	308	4	+	+	CCONJ
cbi-57	308	5	(	(	PUNCT
cbi-57	308	6	1	1	NUM
cbi-57	308	7	+	+	NUM
cbi-57	308	8	2	2	NUM
cbi-57	308	9	+	+	NUM
cbi-57	308	10	3	3	NUM
cbi-57	308	11	+	+	NUM
cbi-57	308	12	.	.	PUNCT
cbi-57	308	13	.	.	PUNCT
cbi-57	308	14	.	.	PUNCT
cbi-57	308	15	)	)	PUNCT
cbi-57	309	1	=	=	PUNCT
cbi-57	309	2	1	1	NUM
cbi-57	309	3	+	+	CCONJ
cbi-57	309	4	(	(	PUNCT
cbi-57	309	5	(	(	PUNCT
cbi-57	309	6	1	1	NUM
cbi-57	309	7	+	+	CCONJ
cbi-57	309	8	#	#	SYM
cbi-57	309	9	1	1	NUM
cbi-57	309	10	)	)	PUNCT
cbi-57	309	11	+	+	CCONJ
cbi-57	309	12	2	2	NUM
cbi-57	309	13	+	+	NUM
cbi-57	309	14	3	3	NUM
cbi-57	309	15	+	+	NUM
cbi-57	309	16	.	.	PUNCT
cbi-57	309	17	.	.	PUNCT
cbi-57	309	18	.	.	PUNCT
cbi-57	309	19	)	)	PUNCT
cbi-57	310	1	=	=	PUNCT
cbi-57	310	2	1	1	NUM
cbi-57	310	3	+	+	CCONJ
cbi-57	310	4	(	(	PUNCT
cbi-57	310	5	0	0	NUM
cbi-57	310	6	+	+	NUM
cbi-57	310	7	2	2	NUM
cbi-57	310	8	+	+	NUM
cbi-57	310	9	3	3	NUM
cbi-57	310	10	+	+	NUM
cbi-57	310	11	.	.	PUNCT
cbi-57	310	12	.	.	PUNCT
cbi-57	310	13	.	.	PUNCT
cbi-57	310	14	)	)	PUNCT
cbi-57	311	1	(	(	PUNCT
cbi-57	311	2	38	38	NUM
cbi-57	311	3	)	)	PUNCT
cbi-57	311	4	8.3	8.3	NUM
cbi-57	311	5	finite	finite	ADJ
cbi-57	311	6	term	term	NOUN
cbi-57	311	7	rearrangement	rearrangement	NOUN
cbi-57	311	8	as	as	SCONJ
cbi-57	311	9	can	can	AUX
cbi-57	311	10	be	be	AUX
cbi-57	311	11	deduced	deduce	VERB
cbi-57	311	12	from	from	ADP
cbi-57	311	13	sections	section	NOUN
cbi-57	311	14	8.1	8.1	NUM
cbi-57	311	15	and	and	CCONJ
cbi-57	311	16	8.2	8.2	NUM
cbi-57	311	17	,	,	PUNCT
cbi-57	311	18	any	any	DET
cbi-57	311	19	number	number	NOUN
cbi-57	311	20	of	of	ADP
cbi-57	311	21	finite	finite	ADJ
cbi-57	311	22	terms	term	NOUN
cbi-57	311	23	in	in	ADP
cbi-57	311	24	a	a	DET
cbi-57	311	25	series	series	NOUN
cbi-57	311	26	can	can	AUX
cbi-57	311	27	be	be	AUX
cbi-57	311	28	rearranged	rearrange	VERB
cbi-57	311	29	in	in	ADP
cbi-57	311	30	position	position	NOUN
cbi-57	311	31	.	.	PUNCT
cbi-57	312	1	that	that	PRON
cbi-57	312	2	is	be	AUX
cbi-57	312	3	,	,	PUNCT
cbi-57	312	4	for	for	ADP
cbi-57	312	5	any	any	DET
cbi-57	312	6	given	give	VERB
cbi-57	312	7	series	series	NOUN
cbi-57	312	8	member	member	NOUN
cbi-57	312	9	with	with	ADP
cbi-57	312	10	a	a	DET
cbi-57	312	11	value	value	NOUN
cbi-57	312	12	of	of	ADP
cbi-57	312	13	a	a	DET
cbi-57	312	14	,	,	PUNCT
cbi-57	312	15	a	a	DET
cbi-57	312	16	#	#	NOUN
cbi-57	312	17	a	a	DET
cbi-57	312	18	can	can	AUX
cbi-57	312	19	be	be	AUX
cbi-57	312	20	added	add	VERB
cbi-57	312	21	to	to	ADP
cbi-57	312	22	the	the	DET
cbi-57	312	23	series	series	NOUN
cbi-57	312	24	,	,	PUNCT
cbi-57	312	25	applying	apply	VERB
cbi-57	312	26	the	the	DET
cbi-57	312	27	#	#	NOUN
cbi-57	312	28	a	a	DET
cbi-57	312	29	such	such	ADJ
cbi-57	312	30	that	that	SCONJ
cbi-57	312	31	it	it	PRON
cbi-57	312	32	cancels	cancel	VERB
cbi-57	312	33	out	out	ADP
cbi-57	312	34	the	the	DET
cbi-57	312	35	value	value	NOUN
cbi-57	312	36	of	of	ADP
cbi-57	312	37	the	the	DET
cbi-57	312	38	series	series	NOUN
cbi-57	312	39	member	member	NOUN
cbi-57	312	40	.	.	PUNCT
cbi-57	313	1	after	after	ADP
cbi-57	313	2	doing	do	VERB
cbi-57	313	3	this	this	PRON
cbi-57	313	4	to	to	ADP
cbi-57	313	5	several	several	ADJ
cbi-57	313	6	series	series	NOUN
cbi-57	313	7	members	member	NOUN
cbi-57	313	8	,	,	PUNCT
cbi-57	313	9	the	the	DET
cbi-57	313	10	inverse	inverse	NOUN
cbi-57	313	11	operations	operation	NOUN
cbi-57	313	12	can	can	AUX
cbi-57	313	13	then	then	ADV
cbi-57	313	14	be	be	AUX
cbi-57	313	15	applied	apply	VERB
cbi-57	313	16	to	to	PART
cbi-57	313	17	move	move	VERB
cbi-57	313	18	these	these	DET
cbi-57	313	19	values	value	NOUN
cbi-57	313	20	to	to	ADP
cbi-57	313	21	any	any	DET
cbi-57	313	22	finite	finite	ADJ
cbi-57	313	23	position	position	NOUN
cbi-57	313	24	in	in	ADP
cbi-57	313	25	the	the	DET
cbi-57	313	26	series	series	NOUN
cbi-57	313	27	.	.	PUNCT
cbi-57	314	1	doing	do	VERB
cbi-57	314	2	this	this	PRON
cbi-57	314	3	will	will	AUX
cbi-57	314	4	preserve	preserve	VERB
cbi-57	314	5	the	the	DET
cbi-57	314	6	partial	partial	ADJ
cbi-57	314	7	summing	sum	VERB
cbi-57	314	8	behavior	behavior	NOUN
cbi-57	314	9	of	of	ADP
cbi-57	314	10	the	the	DET
cbi-57	314	11	series	series	NOUN
cbi-57	314	12	for	for	ADP
cbi-57	314	13	all	all	DET
cbi-57	314	14	partial	partial	ADJ
cbi-57	314	15	sums	sum	NOUN
cbi-57	314	16	after	after	ADP
cbi-57	314	17	the	the	DET
cbi-57	314	18	members	member	NOUN
cbi-57	314	19	which	which	PRON
cbi-57	314	20	have	have	AUX
cbi-57	314	21	been	be	AUX
cbi-57	314	22	manipulated	manipulate	VERB
cbi-57	314	23	.	.	PUNCT
cbi-57	315	1	9	9	NUM
cbi-57	315	2	more	more	ADV
cbi-57	315	3	advanced	advanced	ADJ
cbi-57	315	4	series	series	NOUN
cbi-57	315	5	while	while	SCONJ
cbi-57	315	6	basic	basic	ADJ
cbi-57	315	7	formulas	formula	NOUN
cbi-57	315	8	for	for	ADP
cbi-57	315	9	divergent	divergent	ADJ
cbi-57	315	10	series	series	NOUN
cbi-57	315	11	of	of	ADP
cbi-57	315	12	arithmetic	arithmetic	ADJ
cbi-57	315	13	and	and	CCONJ
cbi-57	315	14	geometric	geometric	ADJ
cbi-57	315	15	series	series	NOUN
cbi-57	315	16	can	can	AUX
cbi-57	315	17	be	be	AUX
cbi-57	315	18	established	establish	VERB
cbi-57	315	19	using	use	VERB
cbi-57	315	20	the	the	DET
cbi-57	315	21	standard	standard	ADJ
cbi-57	315	22	formulas	formula	NOUN
cbi-57	315	23	,	,	PUNCT
cbi-57	315	24	more	more	ADV
cbi-57	315	25	advanced	advanced	ADJ
cbi-57	315	26	series	series	NOUN
cbi-57	315	27	require	require	VERB
cbi-57	315	28	the	the	DET
cbi-57	315	29	use	use	NOUN
cbi-57	315	30	of	of	ADP
cbi-57	315	31	discrete	discrete	ADJ
cbi-57	315	32	integral	integral	ADJ
cbi-57	315	33	calculus7	calculus7	NOUN
cbi-57	315	34	to	to	PART
cbi-57	315	35	establish	establish	VERB
cbi-57	315	36	the	the	DET
cbi-57	315	37	formulas	formula	NOUN
cbi-57	315	38	for	for	ADP
cbi-57	315	39	such	such	ADJ
cbi-57	315	40	series	serie	NOUN
cbi-57	315	41	.	.	PUNCT
cbi-57	316	1	doing	do	VERB
cbi-57	316	2	so	so	ADV
cbi-57	316	3	leads	lead	VERB
cbi-57	316	4	to	to	ADP
cbi-57	316	5	very	very	ADV
cbi-57	316	6	interesting	interesting	ADJ
cbi-57	316	7	results	result	NOUN
cbi-57	316	8	.	.	PUNCT
cbi-57	317	1	9.1	9.1	NUM
cbi-57	317	2	cesàro	cesàro	PROPN
cbi-57	317	3	sums	sum	NOUN
cbi-57	317	4	and	and	CCONJ
cbi-57	317	5	oscillating	oscillate	VERB
cbi-57	317	6	series	series	NOUN
cbi-57	317	7	oscillating	oscillate	VERB
cbi-57	317	8	series	series	NOUN
cbi-57	317	9	have	have	VERB
cbi-57	317	10	an	an	DET
cbi-57	317	11	interesting	interesting	ADJ
cbi-57	317	12	history	history	NOUN
cbi-57	317	13	of	of	ADP
cbi-57	317	14	treatment	treatment	NOUN
cbi-57	317	15	within	within	ADP
cbi-57	317	16	mathematics	mathematic	NOUN
cbi-57	317	17	.	.	PUNCT
cbi-57	318	1	the	the	DET
cbi-57	318	2	standard	standard	ADJ
cbi-57	318	3	series	series	NOUN
cbi-57	318	4	to	to	PART
cbi-57	318	5	consider	consider	VERB
cbi-57	318	6	is	be	AUX
cbi-57	318	7	grandi	grandi	PROPN
cbi-57	318	8	’s	’s	PART
cbi-57	318	9	series	series	NOUN
cbi-57	318	10	:	:	PUNCT
cbi-57	318	11	1#1	1#1	NUM
cbi-57	318	12	+	+	NOUN
cbi-57	318	13	1#1	1#1	NUM
cbi-57	318	14	+	+	NUM
cbi-57	318	15	.	.	PUNCT
cbi-57	318	16	.	.	PUNCT
cbi-57	319	1	..	..	PUNCT
cbi-57	320	1	or	or	CCONJ
cbi-57	320	2	,	,	PUNCT
cbi-57	320	3	written	write	VERB
cbi-57	320	4	more	more	ADV
cbi-57	320	5	formally	formally	ADV
cbi-57	320	6	,	,	PUNCT
cbi-57	320	7	"	"	PUNCT
cbi-57	320	8	!	!	PUNCT
cbi-57	321	1	i=1	i=1	X
cbi-57	321	2	(	(	PUNCT
cbi-57	321	3	#	#	SYM
cbi-57	321	4	1)i+1	1)i+1	NUM
cbi-57	321	5	.	.	PUNCT
cbi-57	322	1	(	(	PUNCT
cbi-57	322	2	39	39	NUM
cbi-57	322	3	)	)	PUNCT
cbi-57	322	4	partials	partial	NOUN
cbi-57	322	5	sums	sum	VERB
cbi-57	322	6	for	for	ADP
cbi-57	322	7	this	this	DET
cbi-57	322	8	series	series	NOUN
cbi-57	322	9	can	can	AUX
cbi-57	322	10	be	be	AUX
cbi-57	322	11	found	find	VERB
cbi-57	322	12	by	by	ADP
cbi-57	322	13	performing	perform	VERB
cbi-57	322	14	a	a	DET
cbi-57	322	15	discrete	discrete	ADJ
cbi-57	322	16	integral	integral	ADJ
cbi-57	322	17	.	.	PUNCT
cbi-57	323	1	n	n	X
cbi-57	323	2	!	!	PUNCT
cbi-57	324	1	i=1	i=1	X
cbi-57	324	2	(	(	PUNCT
cbi-57	324	3	#	#	SYM
cbi-57	324	4	1)i+1	1)i+1	NUM
cbi-57	324	5	=	=	SYM
cbi-57	324	6	1	1	NUM
cbi-57	324	7	2	2	NUM
cbi-57	324	8	(	(	PUNCT
cbi-57	324	9	#	#	SYM
cbi-57	324	10	1)n+1	1)n+1	NUM
cbi-57	325	1	+	+	CCONJ
cbi-57	325	2	1	1	NUM
cbi-57	325	3	2	2	NUM
cbi-57	325	4	.	.	PUNCT
cbi-57	326	1	(	(	PUNCT
cbi-57	326	2	40	40	NUM
cbi-57	326	3	)	)	PUNCT
cbi-57	326	4	what	what	PRON
cbi-57	326	5	is	be	AUX
cbi-57	326	6	particularly	particularly	ADV
cbi-57	326	7	interesting	interesting	ADJ
cbi-57	326	8	about	about	ADP
cbi-57	326	9	this	this	DET
cbi-57	326	10	formula	formula	NOUN
cbi-57	326	11	is	be	AUX
cbi-57	326	12	that	that	SCONJ
cbi-57	326	13	the	the	DET
cbi-57	326	14	cesàro	cesàro	PROPN
cbi-57	326	15	sum	sum	NOUN
cbi-57	326	16	of	of	ADP
cbi-57	326	17	the	the	DET
cbi-57	326	18	infinite	infinite	ADJ
cbi-57	326	19	series	series	NOUN
cbi-57	326	20	(	(	PUNCT
cbi-57	326	21	1	1	NUM
cbi-57	326	22	2	2	NUM
cbi-57	326	23	)	)	PUNCT
cbi-57	326	24	is	be	AUX
cbi-57	326	25	present	present	ADJ
cbi-57	326	26	in	in	ADP
cbi-57	326	27	the	the	DET
cbi-57	326	28	formula	formula	NOUN
cbi-57	326	29	.	.	PUNCT
cbi-57	327	1	now	now	ADV
cbi-57	327	2	,	,	PUNCT
cbi-57	327	3	consider	consider	VERB
cbi-57	327	4	the	the	DET
cbi-57	327	5	oscillating	oscillate	VERB
cbi-57	327	6	series	series	NOUN
cbi-57	327	7	#	#	SYM
cbi-57	327	8	1	1	NUM
cbi-57	327	9	+	+	CCONJ
cbi-57	327	10	1	1	NUM
cbi-57	327	11	#	#	NOUN
cbi-57	327	12	1	1	NUM
cbi-57	327	13	+	+	CCONJ
cbi-57	327	14	.	.	PUNCT
cbi-57	327	15	.	.	PUNCT
cbi-57	328	1	..	..	PUNCT
cbi-57	329	1	this	this	DET
cbi-57	329	2	series	series	NOUN
cbi-57	329	3	has	have	VERB
cbi-57	329	4	the	the	DET
cbi-57	329	5	formula	formula	NOUN
cbi-57	329	6	"	"	PUNCT
cbi-57	329	7	!	!	PUNCT
cbi-57	330	1	i=1	i=1	X
cbi-57	330	2	(	(	PUNCT
cbi-57	330	3	#	#	SYM
cbi-57	330	4	1)i	1)i	NUM
cbi-57	330	5	.	.	PUNCT
cbi-57	331	1	(	(	PUNCT
cbi-57	331	2	41	41	NUM
cbi-57	331	3	)	)	PUNCT
cbi-57	331	4	a	a	DET
cbi-57	331	5	discrete	discrete	ADJ
cbi-57	331	6	integral	integral	NOUN
cbi-57	331	7	of	of	ADP
cbi-57	331	8	the	the	DET
cbi-57	331	9	partial	partial	ADJ
cbi-57	331	10	sums	sum	NOUN
cbi-57	331	11	yields	yield	VERB
cbi-57	331	12	the	the	DET
cbi-57	331	13	formula	formula	NOUN
cbi-57	331	14	n	n	NOUN
cbi-57	331	15	!	!	PUNCT
cbi-57	332	1	i=1	i=1	X
cbi-57	333	1	(	(	PUNCT
cbi-57	333	2	#	#	SYM
cbi-57	333	3	1)i	1)i	NUM
cbi-57	333	4	=	=	SYM
cbi-57	333	5	1	1	NUM
cbi-57	333	6	2	2	NUM
cbi-57	333	7	(	(	PUNCT
cbi-57	333	8	#	#	NOUN
cbi-57	333	9	1)n	1)n	NUM
cbi-57	333	10	#	#	NOUN
cbi-57	333	11	1	1	NUM
cbi-57	333	12	2	2	NUM
cbi-57	333	13	.	.	PUNCT
cbi-57	334	1	(	(	PUNCT
cbi-57	334	2	42	42	X
cbi-57	334	3	)	)	PUNCT
cbi-57	334	4	note	note	VERB
cbi-57	334	5	that	that	SCONJ
cbi-57	334	6	in	in	ADP
cbi-57	334	7	this	this	PRON
cbi-57	334	8	as	as	ADV
cbi-57	334	9	well	well	ADV
cbi-57	334	10	,	,	PUNCT
cbi-57	334	11	#	#	NOUN
cbi-57	334	12	1	1	NUM
cbi-57	334	13	2	2	NUM
cbi-57	334	14	is	be	AUX
cbi-57	334	15	the	the	DET
cbi-57	334	16	cesáro	cesáro	NOUN
cbi-57	334	17	summation	summation	NOUN
cbi-57	334	18	of	of	ADP
cbi-57	334	19	the	the	DET
cbi-57	334	20	infinite	infinite	ADJ
cbi-57	334	21	series	series	NOUN
cbi-57	334	22	.	.	PUNCT
cbi-57	335	1	this	this	PRON
cbi-57	335	2	leads	lead	VERB
cbi-57	335	3	to	to	ADP
cbi-57	335	4	the	the	DET
cbi-57	335	5	conjecture	conjecture	NOUN
cbi-57	335	6	that	that	SCONJ
cbi-57	335	7	,	,	PUNCT
cbi-57	335	8	in	in	ADP
cbi-57	335	9	evaluating	evaluate	VERB
cbi-57	335	10	infinite	infinite	ADJ
cbi-57	335	11	series	series	NOUN
cbi-57	335	12	using	use	VERB
cbi-57	335	13	integral	integral	ADJ
cbi-57	335	14	formulas	formula	NOUN
cbi-57	335	15	,	,	PUNCT
cbi-57	335	16	(	(	PUNCT
cbi-57	335	17	#	#	SYM
cbi-57	335	18	1	1	NUM
cbi-57	335	19	)	)	PUNCT
cbi-57	335	20	"	"	PUNCT
cbi-57	336	1	=	=	SYM
cbi-57	336	2	0	0	NUM
cbi-57	336	3	,	,	PUNCT
cbi-57	336	4	(	(	PUNCT
cbi-57	336	5	43	43	NUM
cbi-57	336	6	)	)	PUNCT
cbi-57	336	7	at	at	ADP
cbi-57	336	8	least	least	ADJ
cbi-57	336	9	for	for	ADP
cbi-57	336	10	additive	additive	ADJ
cbi-57	336	11	o!sets	o!set	NOUN
cbi-57	336	12	of	of	ADP
cbi-57	336	13	!	!	PUNCT
cbi-57	336	14	.	.	PUNCT
cbi-57	337	1	for	for	ADP
cbi-57	337	2	instance	instance	NOUN
cbi-57	337	3	,	,	PUNCT
cbi-57	337	4	in	in	ADP
cbi-57	337	5	the	the	DET
cbi-57	337	6	case	case	NOUN
cbi-57	337	7	of	of	ADP
cbi-57	337	8	grandi	grandi	PROPN
cbi-57	337	9	’s	’s	PART
cbi-57	337	10	series	series	NOUN
cbi-57	337	11	,	,	PUNCT
cbi-57	337	12	using	use	VERB
cbi-57	337	13	the	the	DET
cbi-57	337	14	!	!	PUNCT
cbi-57	337	15	notation	notation	NOUN
cbi-57	337	16	,	,	PUNCT
cbi-57	337	17	the	the	DET
cbi-57	337	18	infinite	infinite	ADJ
cbi-57	337	19	series	series	NOUN
cbi-57	337	20	would	would	AUX
cbi-57	337	21	include	include	VERB
cbi-57	337	22	(	(	PUNCT
cbi-57	337	23	#	#	SYM
cbi-57	337	24	1)!+1	1)!+1	NOUN
cbi-57	337	25	.	.	PUNCT
cbi-57	338	1	the	the	DET
cbi-57	338	2	other	other	ADJ
cbi-57	338	3	series	series	NOUN
cbi-57	338	4	includes	include	VERB
cbi-57	338	5	(	(	PUNCT
cbi-57	338	6	#	#	SYM
cbi-57	338	7	1	1	NUM
cbi-57	338	8	)	)	PUNCT
cbi-57	338	9	!	!	PUNCT
cbi-57	338	10	.	.	PUNCT
cbi-57	339	1	according	accord	VERB
cbi-57	339	2	to	to	ADP
cbi-57	339	3	the	the	DET
cbi-57	339	4	present	present	ADJ
cbi-57	339	5	conjecture	conjecture	NOUN
cbi-57	339	6	,	,	PUNCT
cbi-57	339	7	both	both	PRON
cbi-57	339	8	of	of	ADP
cbi-57	339	9	these	these	DET
cbi-57	339	10	simplify	simplify	ADJ
cbi-57	339	11	7also	7also	NUM
cbi-57	339	12	known	know	VERB
cbi-57	339	13	as	as	ADP
cbi-57	339	14	symbolic	symbolic	ADJ
cbi-57	339	15	summation	summation	NOUN
cbi-57	339	16	.	.	PUNCT
cbi-57	340	1	see	see	VERB
cbi-57	340	2	,	,	PUNCT
cbi-57	340	3	for	for	ADP
cbi-57	340	4	instance	instance	NOUN
cbi-57	340	5	,	,	PUNCT
cbi-57	340	6	chapter	chapter	NOUN
cbi-57	340	7	2	2	NUM
cbi-57	340	8	of	of	ADP
cbi-57	340	9	graham	graham	PROPN
cbi-57	340	10	,	,	PUNCT
cbi-57	340	11	knuth	knuth	PROPN
cbi-57	340	12	,	,	PUNCT
cbi-57	340	13	and	and	CCONJ
cbi-57	340	14	patashnik	patashnik	X
cbi-57	340	15	(	(	PUNCT
cbi-57	340	16	1994	1994	NUM
cbi-57	340	17	)	)	PUNCT
cbi-57	340	18	.	.	PUNCT
cbi-57	341	1	volume	volume	NOUN
cbi-57	341	2	2	2	NUM
cbi-57	341	3	,	,	PUNCT
cbi-57	341	4	issue	issue	NOUN
cbi-57	341	5	1	1	NUM
cbi-57	341	6	9	9	NUM
cbi-57	341	7	.	.	PUNCT
cbi-57	341	8	more	more	ADV
cbi-57	341	9	advanced	advanced	ADJ
cbi-57	341	10	series	series	NOUN
cbi-57	341	11	13	13	NUM
cbi-57	341	12	to	to	PART
cbi-57	341	13	0	0	NUM
cbi-57	341	14	,	,	PUNCT
cbi-57	341	15	at	at	ADP
cbi-57	341	16	least	least	ADJ
cbi-57	341	17	for	for	ADP
cbi-57	341	18	the	the	DET
cbi-57	341	19	purpose	purpose	NOUN
cbi-57	341	20	of	of	ADP
cbi-57	341	21	creating	create	VERB
cbi-57	341	22	formulas	formula	NOUN
cbi-57	341	23	for	for	ADP
cbi-57	341	24	infinite	infinite	ADJ
cbi-57	341	25	series	series	NOUN
cbi-57	341	26	based	base	VERB
cbi-57	341	27	on	on	ADP
cbi-57	341	28	partial	partial	ADJ
cbi-57	341	29	sums	sum	NOUN
cbi-57	341	30	.	.	PUNCT
cbi-57	342	1	this	this	PRON
cbi-57	342	2	can	can	AUX
cbi-57	342	3	be	be	AUX
cbi-57	342	4	understood	understand	VERB
cbi-57	342	5	probabilistically	probabilistically	ADV
cbi-57	342	6	.	.	PUNCT
cbi-57	343	1	since	since	SCONJ
cbi-57	343	2	we	we	PRON
cbi-57	343	3	have	have	VERB
cbi-57	343	4	no	no	DET
cbi-57	343	5	information	information	NOUN
cbi-57	343	6	about	about	ADP
cbi-57	343	7	what	what	PRON
cbi-57	343	8	sign	sign	VERB
cbi-57	343	9	#	#	SYM
cbi-57	343	10	1	1	NUM
cbi-57	343	11	!	!	PUNCT
cbi-57	343	12	will	will	AUX
cbi-57	343	13	have	have	VERB
cbi-57	343	14	,	,	PUNCT
cbi-57	343	15	we	we	PRON
cbi-57	343	16	can	can	AUX
cbi-57	343	17	say	say	VERB
cbi-57	343	18	that	that	SCONJ
cbi-57	343	19	#	#	SYM
cbi-57	343	20	1	1	NUM
cbi-57	343	21	!	!	PUNCT
cbi-57	344	1	=	=	PRON
cbi-57	344	2	±1	±1	VERB
cbi-57	344	3	.	.	PUNCT
cbi-57	345	1	(	(	PUNCT
cbi-57	345	2	44	44	NUM
cbi-57	345	3	)	)	PUNCT
cbi-57	345	4	since	since	SCONJ
cbi-57	345	5	both	both	PRON
cbi-57	345	6	of	of	ADP
cbi-57	345	7	these	these	DET
cbi-57	345	8	possibilities	possibility	NOUN
cbi-57	345	9	are	be	AUX
cbi-57	345	10	equally	equally	ADV
cbi-57	345	11	probable	probable	ADJ
cbi-57	345	12	,	,	PUNCT
cbi-57	345	13	the	the	DET
cbi-57	345	14	limit	limit	NOUN
cbi-57	345	15	towards	towards	ADP
cbi-57	345	16	infinity	infinity	NOUN
cbi-57	345	17	resolves	resolve	NOUN
cbi-57	345	18	to	to	ADP
cbi-57	345	19	their	their	PRON
cbi-57	345	20	average	average	ADJ
cbi-57	345	21	,	,	PUNCT
cbi-57	345	22	or	or	CCONJ
cbi-57	345	23	zero	zero	NUM
cbi-57	345	24	.	.	PUNCT
cbi-57	346	1	also	also	ADV
cbi-57	346	2	,	,	PUNCT
cbi-57	346	3	since	since	SCONJ
cbi-57	346	4	we	we	PRON
cbi-57	346	5	have	have	VERB
cbi-57	346	6	no	no	DET
cbi-57	346	7	information	information	NOUN
cbi-57	346	8	about	about	ADP
cbi-57	346	9	the	the	DET
cbi-57	346	10	sign	sign	NOUN
cbi-57	346	11	of	of	ADP
cbi-57	346	12	#	#	SYM
cbi-57	346	13	1	1	NUM
cbi-57	346	14	!	!	PUNCT
cbi-57	346	15	,	,	PUNCT
cbi-57	346	16	we	we	PRON
cbi-57	346	17	have	have	VERB
cbi-57	346	18	equally	equally	ADV
cbi-57	346	19	little	little	ADJ
cbi-57	346	20	information	information	NOUN
cbi-57	346	21	about	about	ADP
cbi-57	346	22	the	the	DET
cbi-57	346	23	sign	sign	NOUN
cbi-57	346	24	of	of	ADP
cbi-57	346	25	#	#	SYM
cbi-57	346	26	1!+1	1!+1	NOUN
cbi-57	346	27	,	,	PUNCT
cbi-57	346	28	or	or	CCONJ
cbi-57	346	29	any	any	DET
cbi-57	346	30	other	other	ADJ
cbi-57	346	31	variation	variation	NOUN
cbi-57	346	32	on	on	ADP
cbi-57	346	33	!	!	PUNCT
cbi-57	347	1	which	which	PRON
cbi-57	347	2	is	be	AUX
cbi-57	347	3	not	not	PART
cbi-57	347	4	biased	bias	VERB
cbi-57	347	5	towards	towards	ADP
cbi-57	347	6	evenness	evenness	NOUN
cbi-57	347	7	(	(	PUNCT
cbi-57	347	8	e.g.	e.g.	ADV
cbi-57	347	9	,	,	PUNCT
cbi-57	347	10	2	2	NUM
cbi-57	347	11	!	!	PUNCT
cbi-57	347	12	)	)	PUNCT
cbi-57	347	13	.	.	PUNCT
cbi-57	348	1	the	the	DET
cbi-57	348	2	expression	expression	NOUN
cbi-57	348	3	#	#	SYM
cbi-57	348	4	1x	1x	NUM
cbi-57	348	5	has	have	VERB
cbi-57	348	6	an	an	DET
cbi-57	348	7	oscillation	oscillation	NOUN
cbi-57	348	8	pattern	pattern	NOUN
cbi-57	348	9	very	very	ADV
cbi-57	348	10	similar	similar	ADJ
cbi-57	348	11	to	to	ADP
cbi-57	348	12	sin(x	sin(x	PROPN
cbi-57	348	13	)	)	PUNCT
cbi-57	348	14	.	.	PUNCT
cbi-57	349	1	since	since	SCONJ
cbi-57	349	2	(	(	PUNCT
cbi-57	349	3	paterson	paterson	NOUN
cbi-57	349	4	,	,	PUNCT
cbi-57	349	5	2018a	2018a	NUM
cbi-57	349	6	)	)	PUNCT
cbi-57	349	7	showed	show	VERB
cbi-57	349	8	that	that	DET
cbi-57	349	9	sin	sin	NOUN
cbi-57	349	10	(	(	PUNCT
cbi-57	349	11	!	!	PUNCT
cbi-57	349	12	)	)	PUNCT
cbi-57	350	1	=	=	PUNCT
cbi-57	350	2	0	0	NUM
cbi-57	351	1	in	in	ADP
cbi-57	351	2	the	the	DET
cbi-57	351	3	surreal	surreal	ADJ
cbi-57	351	4	numbers	number	NOUN
cbi-57	351	5	,	,	PUNCT
cbi-57	351	6	it	it	PRON
cbi-57	351	7	is	be	AUX
cbi-57	351	8	possible	possible	ADJ
cbi-57	351	9	that	that	SCONJ
cbi-57	351	10	a	a	DET
cbi-57	351	11	similar	similar	ADJ
cbi-57	351	12	proof	proof	NOUN
cbi-57	351	13	may	may	AUX
cbi-57	351	14	be	be	AUX
cbi-57	351	15	found	find	VERB
cbi-57	351	16	for	for	ADP
cbi-57	351	17	#	#	SYM
cbi-57	351	18	1	1	NUM
cbi-57	351	19	!	!	PUNCT
cbi-57	352	1	=	=	SYM
cbi-57	352	2	0	0	NUM
cbi-57	352	3	along	along	ADP
cbi-57	352	4	similar	similar	ADJ
cbi-57	352	5	lines	line	NOUN
cbi-57	352	6	in	in	ADP
cbi-57	352	7	the	the	DET
cbi-57	352	8	hyperreals	hyperreal	NOUN
cbi-57	352	9	.	.	PUNCT
cbi-57	353	1	9.2	9.2	NUM
cbi-57	353	2	other	other	ADJ
cbi-57	353	3	oscillatory	oscillatory	ADJ
cbi-57	353	4	behavior	behavior	NOUN
cbi-57	353	5	because	because	SCONJ
cbi-57	353	6	(	(	PUNCT
cbi-57	353	7	a	a	X
cbi-57	353	8	)	)	PUNCT
cbi-57	353	9	discrete	discrete	ADJ
cbi-57	353	10	integration	integration	NOUN
cbi-57	353	11	can	can	AUX
cbi-57	353	12	be	be	AUX
cbi-57	353	13	used	use	VERB
cbi-57	353	14	to	to	PART
cbi-57	353	15	find	find	VERB
cbi-57	353	16	formulas	formula	NOUN
cbi-57	353	17	for	for	ADP
cbi-57	353	18	series	series	NOUN
cbi-57	353	19	involving	involve	VERB
cbi-57	353	20	partial	partial	ADJ
cbi-57	353	21	sums	sum	NOUN
cbi-57	353	22	,	,	PUNCT
cbi-57	353	23	and	and	CCONJ
cbi-57	353	24	(	(	PUNCT
cbi-57	353	25	b	b	X
cbi-57	353	26	)	)	PUNCT
cbi-57	353	27	the	the	DET
cbi-57	353	28	behavior	behavior	NOUN
cbi-57	353	29	of	of	ADP
cbi-57	353	30	(	(	PUNCT
cbi-57	353	31	#	#	SYM
cbi-57	353	32	1	1	NUM
cbi-57	353	33	)	)	PUNCT
cbi-57	353	34	"	"	PUNCT
cbi-57	353	35	(	(	PUNCT
cbi-57	353	36	for	for	ADP
cbi-57	353	37	infinities	infinity	NOUN
cbi-57	353	38	without	without	ADP
cbi-57	353	39	bias	bias	NOUN
cbi-57	353	40	towards	towards	ADP
cbi-57	353	41	evenness	evenness	NOUN
cbi-57	353	42	)	)	PUNCT
cbi-57	353	43	is	be	AUX
cbi-57	353	44	conjectured	conjecture	VERB
cbi-57	353	45	to	to	PART
cbi-57	353	46	be	be	AUX
cbi-57	353	47	zero	zero	NUM
cbi-57	353	48	,	,	PUNCT
cbi-57	353	49	the	the	DET
cbi-57	353	50	behavior	behavior	NOUN
cbi-57	353	51	of	of	ADP
cbi-57	353	52	a	a	DET
cbi-57	353	53	wide	wide	ADJ
cbi-57	353	54	variety	variety	NOUN
cbi-57	353	55	of	of	ADP
cbi-57	353	56	oscillatory	oscillatory	ADJ
cbi-57	353	57	behaviors	behavior	NOUN
cbi-57	353	58	can	can	AUX
cbi-57	353	59	be	be	AUX
cbi-57	353	60	deduced	deduce	VERB
cbi-57	353	61	.	.	PUNCT
cbi-57	354	1	raising	raise	VERB
cbi-57	354	2	#	#	SYM
cbi-57	354	3	1	1	NUM
cbi-57	354	4	to	to	ADP
cbi-57	354	5	the	the	DET
cbi-57	354	6	ith	ith	PROPN
cbi-57	354	7	power	power	NOUN
cbi-57	354	8	can	can	AUX
cbi-57	354	9	produce	produce	VERB
cbi-57	354	10	all	all	DET
cbi-57	354	11	sorts	sort	NOUN
cbi-57	354	12	of	of	ADP
cbi-57	354	13	oscillatory	oscillatory	ADJ
cbi-57	354	14	behavior	behavior	NOUN
cbi-57	354	15	.	.	PUNCT
cbi-57	355	1	as	as	SCONJ
cbi-57	355	2	has	have	AUX
cbi-57	355	3	been	be	AUX
cbi-57	355	4	seen	see	VERB
cbi-57	355	5	with	with	ADP
cbi-57	355	6	grandi	grandi	PROPN
cbi-57	355	7	’s	’s	PART
cbi-57	355	8	series	series	NOUN
cbi-57	355	9	,	,	PUNCT
cbi-57	355	10	this	this	PRON
cbi-57	355	11	can	can	AUX
cbi-57	355	12	produce	produce	VERB
cbi-57	355	13	a	a	DET
cbi-57	355	14	series	series	NOUN
cbi-57	355	15	of	of	ADP
cbi-57	355	16	values	value	NOUN
cbi-57	355	17	that	that	PRON
cbi-57	355	18	go	go	VERB
cbi-57	355	19	back	back	ADV
cbi-57	355	20	-	-	PUNCT
cbi-57	355	21	and	and	CCONJ
cbi-57	355	22	-	-	PUNCT
cbi-57	355	23	forth	forth	NOUN
cbi-57	355	24	across	across	ADP
cbi-57	355	25	a	a	DET
cbi-57	355	26	mean	mean	ADJ
cbi-57	355	27	value	value	NOUN
cbi-57	355	28	(	(	PUNCT
cbi-57	355	29	the	the	DET
cbi-57	355	30	mean	mean	ADJ
cbi-57	355	31	value	value	NOUN
cbi-57	355	32	can	can	AUX
cbi-57	355	33	be	be	AUX
cbi-57	355	34	changed	change	VERB
cbi-57	355	35	by	by	ADP
cbi-57	355	36	adding	add	VERB
cbi-57	355	37	,	,	PUNCT
cbi-57	355	38	and	and	CCONJ
cbi-57	355	39	the	the	DET
cbi-57	355	40	back	back	ADJ
cbi-57	355	41	-	-	PUNCT
cbi-57	355	42	and	and	CCONJ
cbi-57	355	43	-	-	PUNCT
cbi-57	355	44	forth	forth	NOUN
cbi-57	355	45	can	can	AUX
cbi-57	355	46	be	be	AUX
cbi-57	355	47	changed	change	VERB
cbi-57	355	48	by	by	ADP
cbi-57	355	49	multiplying	multiply	VERB
cbi-57	355	50	)	)	PUNCT
cbi-57	355	51	.	.	PUNCT
cbi-57	356	1	however	however	ADV
cbi-57	356	2	,	,	PUNCT
cbi-57	356	3	(	(	PUNCT
cbi-57	356	4	#	#	SYM
cbi-57	356	5	1)i	1)i	NUM
cbi-57	356	6	can	can	AUX
cbi-57	356	7	also	also	ADV
cbi-57	356	8	be	be	AUX
cbi-57	356	9	expanded	expand	VERB
cbi-57	356	10	to	to	PART
cbi-57	356	11	blank	blank	VERB
cbi-57	356	12	out	out	ADP
cbi-57	356	13	members	member	NOUN
cbi-57	356	14	of	of	ADP
cbi-57	356	15	a	a	DET
cbi-57	356	16	series	series	NOUN
cbi-57	356	17	.	.	PUNCT
cbi-57	357	1	for	for	ADP
cbi-57	357	2	instance	instance	NOUN
cbi-57	357	3	,	,	PUNCT
cbi-57	357	4	to	to	PART
cbi-57	357	5	blank	blank	VERB
cbi-57	357	6	out	out	ADP
cbi-57	357	7	every	every	DET
cbi-57	357	8	other	other	ADJ
cbi-57	357	9	member	member	NOUN
cbi-57	357	10	of	of	ADP
cbi-57	357	11	a	a	DET
cbi-57	357	12	series	series	NOUN
cbi-57	357	13	,	,	PUNCT
cbi-57	357	14	the	the	DET
cbi-57	357	15	formula	formula	NOUN
cbi-57	357	16	(	(	PUNCT
cbi-57	357	17	(	(	PUNCT
cbi-57	357	18	#	#	SYM
cbi-57	357	19	1)i	1)i	NUM
cbi-57	357	20	+	+	CCONJ
cbi-57	357	21	1	1	NUM
cbi-57	357	22	)	)	SYM
cbi-57	357	23	2	2	NUM
cbi-57	357	24	(	(	PUNCT
cbi-57	357	25	45	45	NUM
cbi-57	357	26	)	)	PUNCT
cbi-57	357	27	can	can	AUX
cbi-57	357	28	be	be	AUX
cbi-57	357	29	used	use	VERB
cbi-57	357	30	.	.	PUNCT
cbi-57	358	1	this	this	DET
cbi-57	358	2	simplifies	simplifie	NOUN
cbi-57	358	3	to	to	ADP
cbi-57	358	4	1	1	NUM
cbi-57	358	5	where	where	SCONJ
cbi-57	358	6	i	i	PRON
cbi-57	358	7	is	be	AUX
cbi-57	358	8	even	even	ADV
cbi-57	358	9	and	and	CCONJ
cbi-57	358	10	0	0	NUM
cbi-57	358	11	when	when	SCONJ
cbi-57	358	12	i	i	PRON
cbi-57	358	13	is	be	AUX
cbi-57	358	14	odd	odd	ADJ
cbi-57	358	15	.	.	PUNCT
cbi-57	359	1	therefore	therefore	ADV
cbi-57	359	2	,	,	PUNCT
cbi-57	359	3	by	by	ADP
cbi-57	359	4	multiplying	multiply	VERB
cbi-57	359	5	a	a	DET
cbi-57	359	6	given	give	VERB
cbi-57	359	7	formula	formula	NOUN
cbi-57	359	8	by	by	ADP
cbi-57	359	9	(	(	PUNCT
cbi-57	359	10	45	45	NUM
cbi-57	359	11	)	)	PUNCT
cbi-57	359	12	,	,	PUNCT
cbi-57	359	13	odd	odd	ADV
cbi-57	359	14	-	-	PUNCT
cbi-57	359	15	indexed	index	VERB
cbi-57	359	16	terms	term	NOUN
cbi-57	359	17	of	of	ADP
cbi-57	359	18	the	the	DET
cbi-57	359	19	given	give	VERB
cbi-57	359	20	formula	formula	NOUN
cbi-57	359	21	will	will	AUX
cbi-57	359	22	be	be	AUX
cbi-57	359	23	zeroed	zero	VERB
cbi-57	359	24	out	out	ADP
cbi-57	359	25	.	.	PUNCT
cbi-57	360	1	for	for	ADP
cbi-57	360	2	instance	instance	NOUN
cbi-57	360	3	,	,	PUNCT
cbi-57	360	4	take	take	VERB
cbi-57	360	5	the	the	DET
cbi-57	360	6	series	series	NOUN
cbi-57	360	7	1	1	NUM
cbi-57	361	1	+	+	SYM
cbi-57	361	2	2	2	NUM
cbi-57	361	3	+	+	NUM
cbi-57	361	4	3	3	NUM
cbi-57	361	5	+	+	NUM
cbi-57	361	6	.	.	PUNCT
cbi-57	361	7	.	.	PUNCT
cbi-57	362	1	..	..	PUNCT
cbi-57	363	1	this	this	DET
cbi-57	363	2	series	series	NOUN
cbi-57	363	3	can	can	AUX
cbi-57	363	4	be	be	AUX
cbi-57	363	5	converted	convert	VERB
cbi-57	363	6	to	to	ADP
cbi-57	363	7	the	the	DET
cbi-57	363	8	series	series	NOUN
cbi-57	363	9	0	0	NUM
cbi-57	364	1	+	+	PROPN
cbi-57	364	2	2	2	NUM
cbi-57	364	3	+	+	NOUN
cbi-57	364	4	0	0	NUM
cbi-57	364	5	+	+	NOUN
cbi-57	364	6	4	4	NUM
cbi-57	364	7	+	+	NOUN
cbi-57	364	8	0	0	NUM
cbi-57	364	9	+	+	NOUN
cbi-57	364	10	6	6	NUM
cbi-57	364	11	+	+	NUM
cbi-57	364	12	.	.	PUNCT
cbi-57	364	13	.	.	PUNCT
cbi-57	364	14	.	.	PUNCT
cbi-57	365	1	by	by	ADP
cbi-57	365	2	applying	apply	VERB
cbi-57	365	3	(	(	PUNCT
cbi-57	365	4	45	45	NUM
cbi-57	365	5	)	)	PUNCT
cbi-57	365	6	.	.	PUNCT
cbi-57	366	1	this	this	PRON
cbi-57	366	2	gives	give	VERB
cbi-57	366	3	the	the	DET
cbi-57	366	4	series	series	NOUN
cbi-57	366	5	!	!	PUNCT
cbi-57	366	6	!	!	PUNCT
cbi-57	367	1	i=1	i=1	PROPN
cbi-57	368	1	i	i	PRON
cbi-57	368	2	·	·	PUNCT
cbi-57	368	3	'	'	PUNCT
cbi-57	368	4	(	(	PUNCT
cbi-57	368	5	(	(	PUNCT
cbi-57	368	6	#	#	SYM
cbi-57	368	7	1)i	1)i	NUM
cbi-57	368	8	+	+	CCONJ
cbi-57	368	9	1	1	NUM
cbi-57	368	10	)	)	SYM
cbi-57	368	11	2	2	NUM
cbi-57	368	12	(	(	PUNCT
cbi-57	368	13	.	.	PUNCT
cbi-57	369	1	(	(	PUNCT
cbi-57	369	2	46	46	NUM
cbi-57	369	3	)	)	PUNCT
cbi-57	369	4	the	the	DET
cbi-57	369	5	discrete	discrete	ADJ
cbi-57	369	6	integral	integral	ADJ
cbi-57	369	7	yields	yield	NOUN
cbi-57	369	8	n	n	CCONJ
cbi-57	369	9	!	!	PUNCT
cbi-57	370	1	i=1	i=1	PROPN
cbi-57	371	1	i	i	PRON
cbi-57	371	2	·	·	PUNCT
cbi-57	371	3	'	'	PUNCT
cbi-57	371	4	(	(	PUNCT
cbi-57	371	5	(	(	PUNCT
cbi-57	371	6	#	#	SYM
cbi-57	371	7	1)i	1)i	NUM
cbi-57	371	8	+	+	CCONJ
cbi-57	371	9	1	1	NUM
cbi-57	371	10	)	)	SYM
cbi-57	371	11	2	2	NUM
cbi-57	371	12	(	(	PUNCT
cbi-57	371	13	=	=	NOUN
cbi-57	371	14	1	1	NUM
cbi-57	371	15	8	8	NUM
cbi-57	371	16	%	%	NOUN
cbi-57	371	17	2n2	2n2	NUM
cbi-57	371	18	+	+	CCONJ
cbi-57	371	19	2n(#1)n	2n(#1)n	NUM
cbi-57	371	20	+	+	NUM
cbi-57	371	21	2n	2n	NUM
cbi-57	371	22	+	+	CCONJ
cbi-57	371	23	(	(	PUNCT
cbi-57	371	24	#	#	NOUN
cbi-57	371	25	1)n	1)n	NUM
cbi-57	371	26	#	#	NOUN
cbi-57	371	27	1	1	NUM
cbi-57	371	28	&	&	CCONJ
cbi-57	371	29	(	(	PUNCT
cbi-57	371	30	47	47	NUM
cbi-57	371	31	)	)	PUNCT
cbi-57	371	32	when	when	SCONJ
cbi-57	371	33	n	n	ADV
cbi-57	371	34	=	=	PUNCT
cbi-57	371	35	!	!	PUNCT
cbi-57	372	1	the	the	DET
cbi-57	372	2	formula	formula	NOUN
cbi-57	372	3	runs	run	VERB
cbi-57	372	4	into	into	ADP
cbi-57	372	5	a	a	DET
cbi-57	372	6	problem	problem	NOUN
cbi-57	372	7	with	with	ADP
cbi-57	372	8	simplifying	simplify	VERB
cbi-57	372	9	this	this	PRON
cbi-57	372	10	through	through	ADP
cbi-57	372	11	the	the	DET
cbi-57	372	12	conjecture	conjecture	NOUN
cbi-57	372	13	(	(	PUNCT
cbi-57	372	14	43	43	NUM
cbi-57	372	15	)	)	PUNCT
cbi-57	372	16	because	because	SCONJ
cbi-57	372	17	it	it	PRON
cbi-57	372	18	yields	yield	VERB
cbi-57	372	19	an	an	DET
cbi-57	372	20	indeterminate	indeterminate	ADJ
cbi-57	372	21	form	form	NOUN
cbi-57	372	22	.	.	PUNCT
cbi-57	373	1	the	the	DET
cbi-57	373	2	term	term	NOUN
cbi-57	373	3	2n(#1)n	2n(#1)n	NUM
cbi-57	373	4	becomes	become	VERB
cbi-57	373	5	an	an	DET
cbi-57	373	6	indeterminate	indeterminate	ADJ
cbi-57	373	7	form	form	NOUN
cbi-57	373	8	of	of	ADP
cbi-57	373	9	the	the	DET
cbi-57	373	10	type	type	NOUN
cbi-57	373	11	!	!	PUNCT
cbi-57	374	1	·	·	PUNCT
cbi-57	375	1	0	0	X
cbi-57	375	2	.	.	PUNCT
cbi-57	375	3	this	this	PRON
cbi-57	375	4	can	can	AUX
cbi-57	375	5	be	be	AUX
cbi-57	375	6	resolved	resolve	VERB
cbi-57	375	7	,	,	PUNCT
cbi-57	375	8	however	however	ADV
cbi-57	375	9	,	,	PUNCT
cbi-57	375	10	through	through	ADP
cbi-57	375	11	l’hospital	l’hospital	PROPN
cbi-57	375	12	’s	’s	PART
cbi-57	375	13	rule	rule	NOUN
cbi-57	375	14	.	.	PUNCT
cbi-57	376	1	lim	lim	PROPN
cbi-57	376	2	n%	n%	PROPN
cbi-57	376	3	"	"	PUNCT
cbi-57	376	4	2n	2n	NUM
cbi-57	376	5	(	(	PUNCT
cbi-57	376	6	#	#	SYM
cbi-57	376	7	1)#n	1)#n	NOUN
cbi-57	376	8	=	=	SYM
cbi-57	376	9	2	2	NUM
cbi-57	376	10	#	#	NOUN
cbi-57	376	11	ln(#1)(#1)#n	ln(#1)(#1)#n	NOUN
cbi-57	376	12	=	=	NOUN
cbi-57	376	13	#	#	NOUN
cbi-57	376	14	2	2	NUM
cbi-57	376	15	ln(#1	ln(#1	NOUN
cbi-57	376	16	)	)	PUNCT
cbi-57	376	17	(	(	PUNCT
cbi-57	376	18	#	#	NOUN
cbi-57	376	19	1)n	1)n	NUM
cbi-57	376	20	.	.	PUNCT
cbi-57	377	1	(	(	PUNCT
cbi-57	377	2	48	48	NUM
cbi-57	377	3	)	)	PUNCT
cbi-57	377	4	now	now	ADV
cbi-57	377	5	(	(	PUNCT
cbi-57	377	6	43	43	NUM
cbi-57	377	7	)	)	PUNCT
cbi-57	377	8	can	can	AUX
cbi-57	377	9	be	be	AUX
cbi-57	377	10	applied	apply	VERB
cbi-57	377	11	without	without	ADP
cbi-57	377	12	ambiguity	ambiguity	NOUN
cbi-57	377	13	,	,	PUNCT
cbi-57	377	14	simplifying	simplify	VERB
cbi-57	377	15	it	it	PRON
cbi-57	377	16	to	to	ADP
cbi-57	377	17	zero	zero	NUM
cbi-57	377	18	.	.	PUNCT
cbi-57	378	1	therefore	therefore	ADV
cbi-57	378	2	,	,	PUNCT
cbi-57	378	3	for	for	ADP
cbi-57	378	4	n	n	NOUN
cbi-57	378	5	=	=	PUNCT
cbi-57	378	6	!	!	PUNCT
cbi-57	378	7	,	,	PUNCT
cbi-57	378	8	(	(	PUNCT
cbi-57	378	9	47	47	NUM
cbi-57	378	10	)	)	PUNCT
cbi-57	378	11	simplifies	simplifie	NOUN
cbi-57	378	12	to	to	ADP
cbi-57	378	13	n	n	CCONJ
cbi-57	378	14	!	!	PUNCT
cbi-57	379	1	i=1	i=1	X
cbi-57	380	1	i	i	PRON
cbi-57	380	2	·	·	PUNCT
cbi-57	380	3	'	'	PUNCT
cbi-57	380	4	(	(	PUNCT
cbi-57	380	5	(	(	PUNCT
cbi-57	380	6	#	#	SYM
cbi-57	380	7	1)i	1)i	NUM
cbi-57	380	8	+	+	CCONJ
cbi-57	380	9	1	1	NUM
cbi-57	380	10	)	)	SYM
cbi-57	380	11	2	2	NUM
cbi-57	380	12	(	(	PUNCT
cbi-57	380	13	=	=	NOUN
cbi-57	380	14	1	1	NUM
cbi-57	380	15	8	8	NUM
cbi-57	380	16	%	%	NOUN
cbi-57	380	17	2n2	2n2	NUM
cbi-57	380	18	+	+	CCONJ
cbi-57	380	19	2n	2n	NUM
cbi-57	380	20	#	#	NOUN
cbi-57	380	21	1	1	NUM
cbi-57	380	22	&	&	CCONJ
cbi-57	380	23	.	.	PUNCT
cbi-57	381	1	(	(	PUNCT
cbi-57	381	2	49	49	NUM
cbi-57	381	3	)	)	PUNCT
cbi-57	381	4	this	this	PRON
cbi-57	381	5	means	mean	VERB
cbi-57	381	6	that	that	SCONJ
cbi-57	381	7	the	the	DET
cbi-57	381	8	value	value	NOUN
cbi-57	381	9	of	of	ADP
cbi-57	381	10	this	this	DET
cbi-57	381	11	sum	sum	NOUN
cbi-57	381	12	in	in	ADP
cbi-57	381	13	the	the	DET
cbi-57	381	14	hyperreals	hyperreal	NOUN
cbi-57	381	15	is	be	AUX
cbi-57	381	16	1	1	NUM
cbi-57	381	17	4	4	NUM
cbi-57	381	18	!	!	ADP
cbi-57	381	19	2	2	NUM
cbi-57	382	1	+	+	CCONJ
cbi-57	382	2	1	1	NUM
cbi-57	382	3	4	4	NUM
cbi-57	382	4	!	!	PUNCT
cbi-57	382	5	#	#	NOUN
cbi-57	382	6	1	1	NUM
cbi-57	382	7	8	8	NUM
cbi-57	382	8	$	$	SYM
cbi-57	382	9	1	1	NUM
cbi-57	382	10	4	4	NUM
cbi-57	382	11	!	!	ADP
cbi-57	383	1	2	2	NUM
cbi-57	383	2	.	.	X
cbi-57	383	3	interestingly	interestingly	ADV
cbi-57	383	4	,	,	PUNCT
cbi-57	383	5	this	this	PRON
cbi-57	383	6	is	be	AUX
cbi-57	383	7	a	a	DET
cbi-57	383	8	di!erent	di!erent	NOUN
cbi-57	383	9	result	result	NOUN
cbi-57	383	10	than	than	ADP
cbi-57	383	11	for	for	ADP
cbi-57	383	12	the	the	DET
cbi-57	383	13	simple	simple	ADJ
cbi-57	383	14	series	series	NOUN
cbi-57	383	15	2	2	NUM
cbi-57	383	16	+	+	NOUN
cbi-57	383	17	4	4	NUM
cbi-57	383	18	+	+	NOUN
cbi-57	383	19	6	6	NUM
cbi-57	383	20	+	+	NUM
cbi-57	383	21	.	.	PUNCT
cbi-57	383	22	.	.	PUNCT
cbi-57	384	1	..	..	PUNCT
cbi-57	385	1	since	since	SCONJ
cbi-57	385	2	2	2	NUM
cbi-57	385	3	+	+	SYM
cbi-57	385	4	4	4	NUM
cbi-57	385	5	+	+	NOUN
cbi-57	385	6	6	6	NUM
cbi-57	385	7	+	+	NUM
cbi-57	385	8	.	.	PUNCT
cbi-57	385	9	.	.	PUNCT
cbi-57	385	10	.	.	PUNCT
cbi-57	385	11	is	be	AUX
cbi-57	385	12	a	a	DET
cbi-57	385	13	simple	simple	ADJ
cbi-57	385	14	arithmetic	arithmetic	ADJ
cbi-57	385	15	series	series	NOUN
cbi-57	385	16	,	,	PUNCT
cbi-57	385	17	we	we	PRON
cbi-57	385	18	can	can	AUX
cbi-57	385	19	determine	determine	VERB
cbi-57	385	20	the	the	DET
cbi-57	385	21	hyperreal	hyperreal	ADJ
cbi-57	385	22	sum	sum	NOUN
cbi-57	385	23	using	use	VERB
cbi-57	385	24	(	(	PUNCT
cbi-57	385	25	7	7	NUM
cbi-57	385	26	)	)	PUNCT
cbi-57	385	27	.	.	PUNCT
cbi-57	385	28	!	!	PUNCT
cbi-57	385	29	!	!	PUNCT
cbi-57	386	1	i=1	i=1	PROPN
cbi-57	386	2	2	2	NUM
cbi-57	387	1	+	+	CCONJ
cbi-57	387	2	(	(	PUNCT
cbi-57	387	3	i	i	NOUN
cbi-57	387	4	#	#	NOUN
cbi-57	387	5	1)2	1)2	NUM
cbi-57	387	6	=	=	PUNCT
cbi-57	387	7	!	!	PUNCT
cbi-57	387	8	2	2	NUM
cbi-57	388	1	+	+	CCONJ
cbi-57	388	2	!	!	PUNCT
cbi-57	389	1	$	$	X
cbi-57	389	2	!	!	NOUN
cbi-57	389	3	2	2	NUM
cbi-57	389	4	.	.	PUNCT
cbi-57	390	1	(	(	PUNCT
cbi-57	390	2	50	50	NUM
cbi-57	390	3	)	)	PUNCT
cbi-57	390	4	this	this	PRON
cbi-57	390	5	is	be	AUX
cbi-57	390	6	a	a	DET
cbi-57	390	7	di!erent	di!erent	NOUN
cbi-57	390	8	result	result	NOUN
cbi-57	390	9	than	than	ADP
cbi-57	390	10	what	what	PRON
cbi-57	390	11	was	be	AUX
cbi-57	390	12	obtained	obtain	VERB
cbi-57	390	13	for	for	ADP
cbi-57	390	14	0	0	NUM
cbi-57	390	15	+	+	NUM
cbi-57	390	16	2	2	NUM
cbi-57	390	17	+	+	NUM
cbi-57	390	18	0	0	NUM
cbi-57	391	1	+	+	CCONJ
cbi-57	391	2	4	4	NUM
cbi-57	391	3	+	+	SYM
cbi-57	391	4	0	0	NUM
cbi-57	392	1	+	+	CCONJ
cbi-57	392	2	6	6	NUM
cbi-57	392	3	+	+	CCONJ
cbi-57	392	4	.	.	PUNCT
cbi-57	392	5	.	.	PUNCT
cbi-57	393	1	.	.	PUNCT
cbi-57	393	2	,	,	PUNCT
cbi-57	394	1	which	which	PRON
cbi-57	394	2	was	be	AUX
cbi-57	394	3	1	1	NUM
cbi-57	394	4	4	4	NUM
cbi-57	394	5	!	!	SYM
cbi-57	394	6	2	2	NUM
cbi-57	394	7	,	,	PUNCT
cbi-57	394	8	indicating	indicate	VERB
cbi-57	394	9	that	that	SCONJ
cbi-57	394	10	the	the	DET
cbi-57	394	11	two	two	NUM
cbi-57	394	12	series	series	NOUN
cbi-57	394	13	have	have	VERB
cbi-57	394	14	di!erent	di!erent	NOUN
cbi-57	394	15	behaviors	behavior	NOUN
cbi-57	394	16	.	.	PUNCT
cbi-57	395	1	9.3	9.3	NUM
cbi-57	395	2	1	1	NUM
cbi-57	395	3	#	#	NOUN
cbi-57	395	4	2	2	NUM
cbi-57	395	5	+	+	CCONJ
cbi-57	395	6	3	3	NUM
cbi-57	395	7	#	#	NOUN
cbi-57	395	8	4	4	NUM
cbi-57	395	9	+	+	PUNCT
cbi-57	395	10	.	.	PUNCT
cbi-57	395	11	.	.	PUNCT
cbi-57	395	12	.	.	PUNCT
cbi-57	396	1	euler	euler	PROPN
cbi-57	396	2	’s	’s	PART
cbi-57	396	3	sum	sum	NOUN
cbi-57	396	4	for	for	ADP
cbi-57	396	5	the	the	DET
cbi-57	396	6	series	series	NOUN
cbi-57	396	7	1#2	1#2	NUM
cbi-57	396	8	+	+	NOUN
cbi-57	396	9	3#4	3#4	NUM
cbi-57	396	10	+	+	NUM
cbi-57	396	11	.	.	PUNCT
cbi-57	396	12	.	.	PUNCT
cbi-57	396	13	.	.	PUNCT
cbi-57	397	1	can	can	AUX
cbi-57	397	2	be	be	AUX
cbi-57	397	3	confirmed	confirm	VERB
cbi-57	397	4	using	use	VERB
cbi-57	397	5	this	this	DET
cbi-57	397	6	method	method	NOUN
cbi-57	397	7	as	as	ADV
cbi-57	397	8	well	well	ADV
cbi-57	397	9	.	.	PUNCT
cbi-57	398	1	this	this	DET
cbi-57	398	2	series	series	NOUN
cbi-57	398	3	can	can	AUX
cbi-57	398	4	be	be	AUX
cbi-57	398	5	given	give	VERB
cbi-57	398	6	the	the	DET
cbi-57	398	7	value	value	NOUN
cbi-57	398	8	n	n	NOUN
cbi-57	398	9	!	!	PUNCT
cbi-57	399	1	i=1	i=1	X
cbi-57	399	2	i(#1)i#1	i(#1)i#1	NOUN
cbi-57	400	1	=	=	SYM
cbi-57	400	2	1	1	NUM
cbi-57	400	3	4	4	NUM
cbi-57	400	4	%	%	NOUN
cbi-57	400	5	#	#	SYM
cbi-57	400	6	2n(#1)n	2n(#1)n	NUM
cbi-57	400	7	+	+	CCONJ
cbi-57	400	8	(	(	PUNCT
cbi-57	400	9	#	#	SYM
cbi-57	400	10	1)n+1	1)n+1	NUM
cbi-57	400	11	+	+	NUM
cbi-57	400	12	1	1	NUM
cbi-57	400	13	&	&	CCONJ
cbi-57	400	14	.	.	PUNCT
cbi-57	401	1	(	(	PUNCT
cbi-57	401	2	51	51	NUM
cbi-57	401	3	)	)	PUNCT
cbi-57	401	4	using	use	VERB
cbi-57	401	5	(	(	PUNCT
cbi-57	401	6	43	43	NUM
cbi-57	401	7	)	)	PUNCT
cbi-57	401	8	and	and	CCONJ
cbi-57	401	9	(	(	PUNCT
cbi-57	401	10	48	48	NUM
cbi-57	401	11	)	)	PUNCT
cbi-57	401	12	this	this	DET
cbi-57	401	13	simplifies	simplifie	NOUN
cbi-57	401	14	to	to	ADP
cbi-57	401	15	1	1	NUM
cbi-57	401	16	4	4	NUM
cbi-57	401	17	.	.	PUNCT
cbi-57	402	1	interestingly	interestingly	ADV
cbi-57	402	2	,	,	PUNCT
cbi-57	402	3	this	this	PRON
cbi-57	402	4	is	be	AUX
cbi-57	402	5	a	a	DET
cbi-57	402	6	series	series	NOUN
cbi-57	402	7	that	that	PRON
cbi-57	402	8	is	be	AUX
cbi-57	402	9	not	not	PART
cbi-57	402	10	changed	change	VERB
cbi-57	402	11	even	even	ADV
cbi-57	402	12	in	in	ADP
cbi-57	402	13	its	its	PRON
cbi-57	402	14	exact	exact	ADJ
cbi-57	402	15	hyperreal	hyperreal	NOUN
cbi-57	402	16	by	by	ADP
cbi-57	402	17	prepending	prepende	VERB
cbi-57	402	18	a	a	DET
cbi-57	402	19	zero	zero	NUM
cbi-57	402	20	to	to	ADP
cbi-57	402	21	the	the	DET
cbi-57	402	22	function	function	NOUN
cbi-57	402	23	.	.	PUNCT
cbi-57	403	1	n	n	X
cbi-57	403	2	!	!	PUNCT
cbi-57	404	1	i=1	i=1	PROPN
cbi-57	405	1	(	(	PUNCT
cbi-57	405	2	i	i	NOUN
cbi-57	405	3	#	#	NOUN
cbi-57	405	4	1)(#1)i	1)(#1)i	NUM
cbi-57	405	5	=	=	SYM
cbi-57	405	6	1	1	NUM
cbi-57	405	7	4	4	NUM
cbi-57	405	8	%	%	NOUN
cbi-57	405	9	2n(#1)n	2n(#1)n	NUM
cbi-57	405	10	+	+	CCONJ
cbi-57	405	11	(	(	PUNCT
cbi-57	405	12	#	#	SYM
cbi-57	405	13	1)n+1	1)n+1	NUM
cbi-57	405	14	+	+	NUM
cbi-57	405	15	1	1	NUM
cbi-57	405	16	&	&	CCONJ
cbi-57	405	17	.	.	PUNCT
cbi-57	406	1	(	(	PUNCT
cbi-57	406	2	52	52	NUM
cbi-57	406	3	)	)	PUNCT
cbi-57	406	4	likewise	likewise	ADV
cbi-57	406	5	,	,	PUNCT
cbi-57	406	6	(	(	PUNCT
cbi-57	406	7	43	43	NUM
cbi-57	406	8	)	)	PUNCT
cbi-57	406	9	allows	allow	VERB
cbi-57	406	10	this	this	PRON
cbi-57	406	11	to	to	PART
cbi-57	406	12	reduce	reduce	VERB
cbi-57	406	13	to	to	ADP
cbi-57	406	14	1	1	NUM
cbi-57	406	15	4	4	NUM
cbi-57	406	16	.	.	PUNCT
cbi-57	406	17	14	14	NUM
cbi-57	406	18	hyperreal	hyperreal	ADJ
cbi-57	406	19	numbers	number	NOUN
cbi-57	406	20	for	for	ADP
cbi-57	406	21	infinite	infinite	ADJ
cbi-57	406	22	divergent	divergent	ADJ
cbi-57	406	23	series	series	NOUN
cbi-57	406	24	10	10	NUM
cbi-57	406	25	whole	whole	ADJ
cbi-57	406	26	series	series	NOUN
cbi-57	406	27	manipulation	manipulation	NOUN
cbi-57	406	28	rules	rule	VERB
cbi-57	406	29	in	in	ADP
cbi-57	406	30	addition	addition	NOUN
cbi-57	406	31	to	to	ADP
cbi-57	406	32	manipulation	manipulation	NOUN
cbi-57	406	33	of	of	ADP
cbi-57	406	34	finite	finite	ADJ
cbi-57	406	35	partial	partial	ADJ
cbi-57	406	36	sums	sum	NOUN
cbi-57	406	37	of	of	ADP
cbi-57	406	38	a	a	DET
cbi-57	406	39	series	series	NOUN
cbi-57	406	40	,	,	PUNCT
cbi-57	406	41	certain	certain	ADJ
cbi-57	406	42	operations	operation	NOUN
cbi-57	406	43	can	can	AUX
cbi-57	406	44	(	(	PUNCT
cbi-57	406	45	and	and	CCONJ
cbi-57	406	46	ca	can	AUX
cbi-57	406	47	n’t	not	PART
cbi-57	406	48	)	)	PUNCT
cbi-57	406	49	be	be	AUX
cbi-57	406	50	performed	perform	VERB
cbi-57	406	51	to	to	ADP
cbi-57	406	52	the	the	DET
cbi-57	406	53	series	series	NOUN
cbi-57	406	54	as	as	ADP
cbi-57	406	55	a	a	DET
cbi-57	406	56	whole	whole	NOUN
cbi-57	406	57	.	.	PUNCT
cbi-57	407	1	in	in	ADP
cbi-57	407	2	this	this	DET
cbi-57	407	3	section	section	NOUN
cbi-57	407	4	,	,	PUNCT
cbi-57	407	5	some	some	PRON
cbi-57	407	6	of	of	ADP
cbi-57	407	7	these	these	DET
cbi-57	407	8	operations	operation	NOUN
cbi-57	407	9	will	will	AUX
cbi-57	407	10	be	be	AUX
cbi-57	407	11	considered	consider	VERB
cbi-57	407	12	.	.	PUNCT
cbi-57	408	1	10.1	10.1	NUM
cbi-57	408	2	scalar	scalar	ADJ
cbi-57	408	3	multiplication	multiplication	NOUN
cbi-57	408	4	because	because	SCONJ
cbi-57	408	5	of	of	ADP
cbi-57	408	6	the	the	DET
cbi-57	408	7	distributivity	distributivity	NOUN
cbi-57	408	8	of	of	ADP
cbi-57	408	9	multiplication	multiplication	NOUN
cbi-57	408	10	,	,	PUNCT
cbi-57	408	11	multiplication	multiplication	NOUN
cbi-57	408	12	of	of	ADP
cbi-57	408	13	a	a	DET
cbi-57	408	14	series	series	NOUN
cbi-57	408	15	by	by	ADP
cbi-57	408	16	a	a	DET
cbi-57	408	17	scalar	scalar	ADJ
cbi-57	408	18	value	value	NOUN
cbi-57	408	19	will	will	AUX
cbi-57	408	20	distribute	distribute	VERB
cbi-57	408	21	the	the	DET
cbi-57	408	22	scalar	scalar	ADJ
cbi-57	408	23	multiplication	multiplication	NOUN
cbi-57	408	24	to	to	ADP
cbi-57	408	25	every	every	DET
cbi-57	408	26	term	term	NOUN
cbi-57	408	27	.	.	PUNCT
cbi-57	409	1	2(1	2(1	NUM
cbi-57	410	1	+	+	CCONJ
cbi-57	410	2	2	2	NUM
cbi-57	410	3	+	+	NUM
cbi-57	410	4	3	3	NUM
cbi-57	410	5	+	+	NUM
cbi-57	410	6	.	.	PUNCT
cbi-57	410	7	.	.	PUNCT
cbi-57	410	8	.	.	PUNCT
cbi-57	410	9	)	)	PUNCT
cbi-57	411	1	=	=	PUNCT
cbi-57	411	2	(	(	PUNCT
cbi-57	411	3	2	2	NUM
cbi-57	411	4	·	·	SYM
cbi-57	411	5	1	1	NUM
cbi-57	412	1	+	+	CCONJ
cbi-57	412	2	2	2	NUM
cbi-57	412	3	·	·	SYM
cbi-57	412	4	2	2	NUM
cbi-57	412	5	+	+	CCONJ
cbi-57	412	6	2	2	NUM
cbi-57	412	7	·	·	SYM
cbi-57	412	8	3	3	NUM
cbi-57	413	1	+	+	CCONJ
cbi-57	413	2	.	.	PUNCT
cbi-57	413	3	.	.	PUNCT
cbi-57	413	4	.	.	PUNCT
cbi-57	413	5	)	)	PUNCT
cbi-57	413	6	.	.	PUNCT
cbi-57	414	1	(	(	PUNCT
cbi-57	414	2	53	53	NUM
cbi-57	414	3	)	)	PUNCT
cbi-57	414	4	or	or	CCONJ
cbi-57	414	5	,	,	PUNCT
cbi-57	414	6	written	write	VERB
cbi-57	414	7	as	as	ADP
cbi-57	414	8	a	a	DET
cbi-57	414	9	formula	formula	NOUN
cbi-57	414	10	,	,	PUNCT
cbi-57	414	11	n	n	PRON
cbi-57	414	12	!	!	PUNCT
cbi-57	414	13	!	!	PUNCT
cbi-57	415	1	i=1	i=1	PROPN
cbi-57	416	1	f	f	X
cbi-57	416	2	(	(	PUNCT
cbi-57	416	3	i	i	NOUN
cbi-57	416	4	)	)	PUNCT
cbi-57	416	5	=	=	PUNCT
cbi-57	416	6	!	!	PUNCT
cbi-57	416	7	!	!	PUNCT
cbi-57	417	1	i=1	i=1	PROPN
cbi-57	418	1	n	n	PRON
cbi-57	418	2	f	f	PROPN
cbi-57	418	3	(	(	PUNCT
cbi-57	418	4	i	i	NOUN
cbi-57	418	5	)	)	PUNCT
cbi-57	418	6	.	.	PUNCT
cbi-57	419	1	(	(	PUNCT
cbi-57	419	2	54	54	NUM
cbi-57	419	3	)	)	PUNCT
cbi-57	419	4	10.2	10.2	NUM
cbi-57	419	5	whole	whole	ADJ
cbi-57	419	6	series	series	NOUN
cbi-57	419	7	addition	addition	NOUN
cbi-57	419	8	adding	add	VERB
cbi-57	419	9	two	two	NUM
cbi-57	419	10	series	series	NOUN
cbi-57	419	11	together	together	ADV
cbi-57	419	12	is	be	AUX
cbi-57	419	13	equivalent	equivalent	ADJ
cbi-57	419	14	to	to	ADP
cbi-57	419	15	a	a	DET
cbi-57	419	16	term	term	NOUN
cbi-57	419	17	-	-	PUNCT
cbi-57	419	18	by	by	ADP
cbi-57	419	19	-	-	PUNCT
cbi-57	419	20	term	term	NOUN
cbi-57	419	21	addition	addition	NOUN
cbi-57	419	22	of	of	ADP
cbi-57	419	23	the	the	DET
cbi-57	419	24	series	series	NOUN
cbi-57	419	25	.	.	PUNCT
cbi-57	420	1	since	since	SCONJ
cbi-57	420	2	the	the	DET
cbi-57	420	3	method	method	NOUN
cbi-57	420	4	presented	present	VERB
cbi-57	420	5	here	here	ADV
cbi-57	420	6	is	be	AUX
cbi-57	420	7	based	base	VERB
cbi-57	420	8	on	on	ADP
cbi-57	420	9	partial	partial	ADJ
cbi-57	420	10	sums	sum	NOUN
cbi-57	420	11	,	,	PUNCT
cbi-57	420	12	term	term	NOUN
cbi-57	420	13	-	-	PUNCT
cbi-57	420	14	by	by	ADP
cbi-57	420	15	-	-	PUNCT
cbi-57	420	16	term	term	NOUN
cbi-57	420	17	addition	addition	NOUN
cbi-57	420	18	only	only	ADV
cbi-57	420	19	works	work	VERB
cbi-57	420	20	when	when	SCONJ
cbi-57	420	21	the	the	DET
cbi-57	420	22	lower	low	ADJ
cbi-57	420	23	and	and	CCONJ
cbi-57	420	24	upper	upper	ADJ
cbi-57	420	25	bounds	bound	NOUN
cbi-57	420	26	of	of	ADP
cbi-57	420	27	the	the	DET
cbi-57	420	28	terms	term	NOUN
cbi-57	420	29	are	be	AUX
cbi-57	420	30	identical	identical	ADJ
cbi-57	420	31	.	.	PUNCT
cbi-57	421	1	therefore	therefore	ADV
cbi-57	421	2	,	,	PUNCT
cbi-57	421	3	!	!	PUNCT
cbi-57	421	4	"	"	PUNCT
cbi-57	421	5	!	!	PUNCT
cbi-57	421	6	!	!	PUNCT
cbi-57	422	1	i=1	i=1	PROPN
cbi-57	423	1	f	f	X
cbi-57	423	2	(	(	PUNCT
cbi-57	423	3	i)#$	i)#$	NOUN
cbi-57	423	4	+	+	CCONJ
cbi-57	423	5	!	!	PUNCT
cbi-57	423	6	"	"	PUNCT
cbi-57	423	7	!	!	PUNCT
cbi-57	423	8	!	!	PUNCT
cbi-57	424	1	i=1	i=1	PROPN
cbi-57	425	1	g(i)#$	g(i)#$	PROPN
cbi-57	425	2	=	=	PUNCT
cbi-57	425	3	!	!	PUNCT
cbi-57	425	4	!	!	PUNCT
cbi-57	426	1	i=1	i=1	PROPN
cbi-57	427	1	f	f	X
cbi-57	427	2	(	(	PUNCT
cbi-57	427	3	i	i	NOUN
cbi-57	427	4	)	)	PUNCT
cbi-57	428	1	+	+	NUM
cbi-57	428	2	g(i	g(i	NOUN
cbi-57	428	3	)	)	PUNCT
cbi-57	428	4	.	.	PUNCT
cbi-57	429	1	(	(	PUNCT
cbi-57	429	2	55	55	NUM
cbi-57	429	3	)	)	PUNCT
cbi-57	429	4	however	however	ADV
cbi-57	429	5	,	,	PUNCT
cbi-57	429	6	!	!	PUNCT
cbi-57	429	7	"	"	PUNCT
cbi-57	429	8	!	!	PUNCT
cbi-57	429	9	!	!	PUNCT
cbi-57	430	1	i=0	i=0	PROPN
cbi-57	430	2	f	f	X
cbi-57	430	3	(	(	PUNCT
cbi-57	430	4	i)#$	i)#$	NOUN
cbi-57	430	5	+	+	CCONJ
cbi-57	430	6	!	!	PUNCT
cbi-57	430	7	"	"	PUNCT
cbi-57	430	8	!	!	PUNCT
cbi-57	430	9	!	!	PUNCT
cbi-57	431	1	i=1	i=1	PROPN
cbi-57	431	2	g(i)#$	g(i)#$	PROPN
cbi-57	431	3	!	!	PUNCT
cbi-57	431	4	!	!	PUNCT
cbi-57	431	5	!	!	PUNCT
cbi-57	432	1	i=1	i=1	PROPN
cbi-57	433	1	f	f	X
cbi-57	433	2	(	(	PUNCT
cbi-57	433	3	i	i	NOUN
cbi-57	433	4	)	)	PUNCT
cbi-57	434	1	+	+	NUM
cbi-57	434	2	g(i	g(i	NOUN
cbi-57	434	3	)	)	PUNCT
cbi-57	434	4	(	(	PUNCT
cbi-57	434	5	56	56	NUM
cbi-57	434	6	)	)	PUNCT
cbi-57	434	7	because	because	SCONJ
cbi-57	434	8	the	the	DET
cbi-57	434	9	limits	limit	NOUN
cbi-57	434	10	of	of	ADP
cbi-57	434	11	summation	summation	NOUN
cbi-57	434	12	di!er	di!er	PROPN
cbi-57	434	13	.	.	PUNCT
cbi-57	435	1	again	again	ADV
cbi-57	435	2	,	,	PUNCT
cbi-57	435	3	to	to	PART
cbi-57	435	4	see	see	VERB
cbi-57	435	5	why	why	SCONJ
cbi-57	435	6	this	this	PRON
cbi-57	435	7	is	be	AUX
cbi-57	435	8	the	the	DET
cbi-57	435	9	case	case	NOUN
cbi-57	435	10	,	,	PUNCT
cbi-57	435	11	imagine	imagine	VERB
cbi-57	435	12	replacing	replace	VERB
cbi-57	435	13	!	!	PUNCT
cbi-57	436	1	with	with	ADP
cbi-57	436	2	a	a	DET
cbi-57	436	3	fixed	fix	VERB
cbi-57	436	4	scalar	scalar	NOUN
cbi-57	436	5	such	such	ADJ
cbi-57	436	6	as	as	ADP
cbi-57	436	7	5	5	NUM
cbi-57	436	8	.	.	PUNCT
cbi-57	437	1	in	in	ADP
cbi-57	437	2	(	(	PUNCT
cbi-57	437	3	56	56	NUM
cbi-57	437	4	)	)	PUNCT
cbi-57	437	5	,	,	PUNCT
cbi-57	437	6	the	the	DET
cbi-57	437	7	left	leave	VERB
cbi-57	437	8	-	-	PUNCT
cbi-57	437	9	hand	hand	NOUN
cbi-57	437	10	addend	addend	NOUN
cbi-57	437	11	would	would	AUX
cbi-57	437	12	have	have	VERB
cbi-57	437	13	a	a	DET
cbi-57	437	14	di!erent	di!erent	NOUN
cbi-57	437	15	number	number	NOUN
cbi-57	437	16	of	of	ADP
cbi-57	437	17	terms	term	NOUN
cbi-57	437	18	than	than	ADP
cbi-57	437	19	the	the	DET
cbi-57	437	20	right	right	ADJ
cbi-57	437	21	-	-	PUNCT
cbi-57	437	22	hand	hand	NOUN
cbi-57	437	23	addend	addend	NOUN
cbi-57	437	24	.	.	PUNCT
cbi-57	438	1	10.3	10.3	NUM
cbi-57	438	2	series	series	NOUN
cbi-57	438	3	spacing	space	VERB
cbi-57	438	4	as	as	SCONJ
cbi-57	438	5	noted	note	VERB
cbi-57	438	6	in	in	ADP
cbi-57	438	7	section	section	NOUN
cbi-57	438	8	8.2	8.2	NUM
cbi-57	438	9	,	,	PUNCT
cbi-57	438	10	adding	add	VERB
cbi-57	438	11	or	or	CCONJ
cbi-57	438	12	removing	remove	VERB
cbi-57	438	13	elements	element	NOUN
cbi-57	438	14	of	of	ADP
cbi-57	438	15	a	a	DET
cbi-57	438	16	series	series	NOUN
cbi-57	438	17	,	,	PUNCT
cbi-57	438	18	even	even	ADV
cbi-57	438	19	if	if	SCONJ
cbi-57	438	20	they	they	PRON
cbi-57	438	21	are	be	AUX
cbi-57	438	22	zero	zero	NUM
cbi-57	438	23	,	,	PUNCT
cbi-57	438	24	has	have	VERB
cbi-57	438	25	an	an	DET
cbi-57	438	26	e!ect	e!ect	PROPN
cbi-57	438	27	on	on	ADP
cbi-57	438	28	the	the	DET
cbi-57	438	29	sum	sum	NOUN
cbi-57	438	30	of	of	ADP
cbi-57	438	31	the	the	DET
cbi-57	438	32	resulting	result	VERB
cbi-57	438	33	series	series	NOUN
cbi-57	438	34	.	.	PUNCT
cbi-57	439	1	this	this	DET
cbi-57	439	2	e!ect	e!ect	PROPN
cbi-57	439	3	can	can	AUX
cbi-57	439	4	be	be	AUX
cbi-57	439	5	calculated	calculate	VERB
cbi-57	439	6	using	use	VERB
cbi-57	439	7	the	the	DET
cbi-57	439	8	considerations	consideration	NOUN
cbi-57	439	9	discussed	discuss	VERB
cbi-57	439	10	in	in	ADP
cbi-57	439	11	section	section	NOUN
cbi-57	439	12	9	9	NUM
cbi-57	439	13	.	.	PUNCT
cbi-57	440	1	for	for	ADP
cbi-57	440	2	instance	instance	NOUN
cbi-57	440	3	,	,	PUNCT
cbi-57	440	4	the	the	DET
cbi-57	440	5	series	series	NOUN
cbi-57	440	6	1	1	NUM
cbi-57	440	7	+	+	SYM
cbi-57	440	8	1	1	NUM
cbi-57	440	9	+	+	NUM
cbi-57	440	10	1	1	NUM
cbi-57	440	11	+	+	NUM
cbi-57	440	12	.	.	PUNCT
cbi-57	440	13	.	.	PUNCT
cbi-57	440	14	.	.	PUNCT
cbi-57	441	1	can	can	AUX
cbi-57	441	2	be	be	AUX
cbi-57	441	3	spaced	space	VERB
cbi-57	441	4	out	out	ADP
cbi-57	441	5	by	by	ADP
cbi-57	441	6	adding	add	VERB
cbi-57	441	7	in	in	ADP
cbi-57	441	8	zeroes	zero	NOUN
cbi-57	441	9	,	,	PUNCT
cbi-57	441	10	to	to	PART
cbi-57	441	11	make	make	VERB
cbi-57	441	12	1	1	NUM
cbi-57	441	13	+	+	CCONJ
cbi-57	441	14	0	0	NUM
cbi-57	442	1	+	+	CCONJ
cbi-57	442	2	1	1	NUM
cbi-57	442	3	+	+	NUM
cbi-57	442	4	0	0	NUM
cbi-57	443	1	+	+	CCONJ
cbi-57	443	2	.	.	PUNCT
cbi-57	443	3	.	.	PUNCT
cbi-57	444	1	..	..	PUNCT
cbi-57	445	1	a	a	DET
cbi-57	445	2	variation	variation	NOUN
cbi-57	445	3	of	of	ADP
cbi-57	445	4	the	the	DET
cbi-57	445	5	oscillatory	oscillatory	ADJ
cbi-57	445	6	pattern	pattern	NOUN
cbi-57	445	7	in	in	ADP
cbi-57	445	8	(	(	PUNCT
cbi-57	445	9	45	45	NUM
cbi-57	445	10	)	)	PUNCT
cbi-57	445	11	can	can	AUX
cbi-57	445	12	be	be	AUX
cbi-57	445	13	used	use	VERB
cbi-57	445	14	to	to	PART
cbi-57	445	15	give	give	VERB
cbi-57	445	16	the	the	DET
cbi-57	445	17	series	series	NOUN
cbi-57	445	18	the	the	DET
cbi-57	445	19	formula	formula	NOUN
cbi-57	445	20	n	n	NOUN
cbi-57	445	21	!	!	PUNCT
cbi-57	446	1	i=1	i=1	X
cbi-57	447	1	(	(	PUNCT
cbi-57	447	2	(	(	PUNCT
cbi-57	447	3	#	#	SYM
cbi-57	447	4	1)i+1	1)i+1	NUM
cbi-57	447	5	+	+	NUM
cbi-57	447	6	1	1	NUM
cbi-57	447	7	)	)	PUNCT
cbi-57	447	8	2	2	NUM
cbi-57	447	9	.	.	PUNCT
cbi-57	448	1	(	(	PUNCT
cbi-57	448	2	57	57	NUM
cbi-57	448	3	)	)	PUNCT
cbi-57	448	4	the	the	DET
cbi-57	448	5	discrete	discrete	ADJ
cbi-57	448	6	integral	integral	NOUN
cbi-57	448	7	of	of	ADP
cbi-57	448	8	this	this	DET
cbi-57	448	9	yields	yield	NOUN
cbi-57	448	10	the	the	DET
cbi-57	448	11	formula	formula	NOUN
cbi-57	448	12	1	1	NUM
cbi-57	448	13	2	2	NUM
cbi-57	448	14	n	n	NOUN
cbi-57	448	15	+	+	CCONJ
cbi-57	448	16	1	1	NUM
cbi-57	448	17	4	4	NUM
cbi-57	448	18	(	(	PUNCT
cbi-57	448	19	#	#	SYM
cbi-57	448	20	1)n+1	1)n+1	NUM
cbi-57	449	1	+	+	CCONJ
cbi-57	449	2	1	1	NUM
cbi-57	449	3	4	4	NUM
cbi-57	449	4	(	(	PUNCT
cbi-57	449	5	58	58	NUM
cbi-57	449	6	)	)	PUNCT
cbi-57	449	7	using	use	VERB
cbi-57	449	8	conjecture	conjecture	NOUN
cbi-57	449	9	(	(	PUNCT
cbi-57	449	10	43	43	NUM
cbi-57	449	11	)	)	PUNCT
cbi-57	449	12	this	this	PRON
cbi-57	449	13	reduces	reduce	VERB
cbi-57	449	14	to	to	ADP
cbi-57	449	15	the	the	DET
cbi-57	449	16	hyperreal	hyperreal	NOUN
cbi-57	449	17	value	value	NOUN
cbi-57	449	18	1	1	NUM
cbi-57	449	19	2	2	NUM
cbi-57	449	20	!	!	PUNCT
cbi-57	450	1	+	+	CCONJ
cbi-57	450	2	1	1	NUM
cbi-57	450	3	4	4	NUM
cbi-57	450	4	$	$	SYM
cbi-57	450	5	1	1	NUM
cbi-57	450	6	2	2	NUM
cbi-57	450	7	!	!	PUNCT
cbi-57	450	8	.	.	PUNCT
cbi-57	451	1	this	this	PRON
cbi-57	451	2	is	be	AUX
cbi-57	451	3	a	a	DET
cbi-57	451	4	slightly	slightly	ADV
cbi-57	451	5	di!erent	di!erent	NOUN
cbi-57	451	6	value	value	NOUN
cbi-57	451	7	(	(	PUNCT
cbi-57	451	8	but	but	CCONJ
cbi-57	451	9	with	with	ADP
cbi-57	451	10	the	the	DET
cbi-57	451	11	same	same	ADJ
cbi-57	451	12	principal	principal	ADJ
cbi-57	451	13	value	value	NOUN
cbi-57	451	14	)	)	PUNCT
cbi-57	451	15	than	than	ADP
cbi-57	451	16	for	for	ADP
cbi-57	451	17	the	the	DET
cbi-57	451	18	series	series	NOUN
cbi-57	451	19	0	0	PUNCT
cbi-57	452	1	+	+	CCONJ
cbi-57	452	2	1	1	NUM
cbi-57	452	3	+	+	SYM
cbi-57	452	4	0	0	NUM
cbi-57	453	1	+	+	NUM
cbi-57	453	2	1	1	NUM
cbi-57	453	3	+	+	NUM
cbi-57	453	4	.	.	PUNCT
cbi-57	453	5	.	.	PUNCT
cbi-57	454	1	..	..	PUNCT
cbi-57	455	1	this	this	DET
cbi-57	455	2	series	series	NOUN
cbi-57	455	3	can	can	AUX
cbi-57	455	4	be	be	AUX
cbi-57	455	5	represented	represent	VERB
cbi-57	455	6	as	as	ADP
cbi-57	455	7	n	n	X
cbi-57	455	8	!	!	PUNCT
cbi-57	456	1	i=1	i=1	X
cbi-57	457	1	(	(	PUNCT
cbi-57	457	2	(	(	PUNCT
cbi-57	457	3	#	#	SYM
cbi-57	457	4	1)i	1)i	NUM
cbi-57	457	5	+	+	CCONJ
cbi-57	457	6	1	1	NUM
cbi-57	457	7	)	)	SYM
cbi-57	457	8	2	2	NUM
cbi-57	457	9	=	=	SYM
cbi-57	457	10	1	1	NUM
cbi-57	457	11	2	2	NUM
cbi-57	457	12	n	n	NOUN
cbi-57	457	13	+	+	CCONJ
cbi-57	457	14	1	1	NUM
cbi-57	457	15	2	2	NUM
cbi-57	457	16	(	(	PUNCT
cbi-57	457	17	#	#	NOUN
cbi-57	457	18	1)n	1)n	NUM
cbi-57	457	19	#	#	NOUN
cbi-57	457	20	1	1	NUM
cbi-57	457	21	4	4	NUM
cbi-57	457	22	.	.	PUNCT
cbi-57	458	1	(	(	PUNCT
cbi-57	458	2	59	59	NUM
cbi-57	458	3	)	)	PUNCT
cbi-57	458	4	using	use	VERB
cbi-57	458	5	conjecture	conjecture	NOUN
cbi-57	458	6	(	(	PUNCT
cbi-57	458	7	43	43	NUM
cbi-57	458	8	)	)	PUNCT
cbi-57	458	9	,	,	PUNCT
cbi-57	458	10	the	the	DET
cbi-57	458	11	hyperreal	hyperreal	NOUN
cbi-57	458	12	value	value	NOUN
cbi-57	458	13	for	for	ADP
cbi-57	458	14	this	this	PRON
cbi-57	458	15	is	be	AUX
cbi-57	458	16	1	1	NUM
cbi-57	458	17	2	2	NUM
cbi-57	458	18	!	!	PUNCT
cbi-57	458	19	#	#	SYM
cbi-57	458	20	1	1	NUM
cbi-57	458	21	4	4	NUM
cbi-57	458	22	$	$	SYM
cbi-57	458	23	1	1	NUM
cbi-57	458	24	2	2	NUM
cbi-57	458	25	!	!	PUNCT
cbi-57	458	26	.	.	PUNCT
cbi-57	459	1	if	if	SCONJ
cbi-57	459	2	(	(	PUNCT
cbi-57	459	3	58	58	NUM
cbi-57	459	4	)	)	PUNCT
cbi-57	459	5	and	and	CCONJ
cbi-57	459	6	(	(	PUNCT
cbi-57	459	7	59	59	NUM
cbi-57	459	8	)	)	PUNCT
cbi-57	459	9	were	be	AUX
cbi-57	459	10	added	add	VERB
cbi-57	459	11	,	,	PUNCT
cbi-57	459	12	it	it	PRON
cbi-57	459	13	should	should	AUX
cbi-57	459	14	be	be	AUX
cbi-57	459	15	equivalent	equivalent	ADJ
cbi-57	459	16	whether	whether	SCONJ
cbi-57	459	17	they	they	PRON
cbi-57	459	18	are	be	AUX
cbi-57	459	19	added	add	VERB
cbi-57	459	20	term	term	NOUN
cbi-57	459	21	-	-	PUNCT
cbi-57	459	22	by	by	ADP
cbi-57	459	23	-	-	PUNCT
cbi-57	459	24	term	term	NOUN
cbi-57	459	25	(	(	PUNCT
cbi-57	459	26	section	section	NOUN
cbi-57	459	27	10.2	10.2	NUM
cbi-57	459	28	)	)	PUNCT
cbi-57	459	29	or	or	CCONJ
cbi-57	459	30	by	by	ADP
cbi-57	459	31	summing	sum	VERB
cbi-57	459	32	their	their	PRON
cbi-57	459	33	relevant	relevant	ADJ
cbi-57	459	34	values	value	NOUN
cbi-57	459	35	.	.	PUNCT
cbi-57	460	1	summing	sum	VERB
cbi-57	460	2	term	term	NOUN
cbi-57	460	3	-	-	PUNCT
cbi-57	460	4	by	by	ADP
cbi-57	460	5	-	-	PUNCT
cbi-57	460	6	term	term	NOUN
cbi-57	460	7	it	it	PRON
cbi-57	460	8	is	be	AUX
cbi-57	460	9	apparent	apparent	ADJ
cbi-57	460	10	that	that	SCONJ
cbi-57	460	11	(	(	PUNCT
cbi-57	460	12	1	1	NUM
cbi-57	460	13	+	+	SYM
cbi-57	460	14	0	0	NUM
cbi-57	460	15	+	+	CCONJ
cbi-57	460	16	1	1	NUM
cbi-57	460	17	+	+	NUM
cbi-57	460	18	0	0	NUM
cbi-57	461	1	+	+	CCONJ
cbi-57	461	2	.	.	PUNCT
cbi-57	461	3	.	.	PUNCT
cbi-57	461	4	.	.	PUNCT
cbi-57	461	5	)	)	PUNCT
cbi-57	462	1	+	+	CCONJ
cbi-57	463	1	(	(	PUNCT
cbi-57	463	2	0	0	NUM
cbi-57	463	3	+	+	NUM
cbi-57	463	4	1	1	NUM
cbi-57	463	5	+	+	SYM
cbi-57	463	6	0	0	NUM
cbi-57	464	1	+	+	NUM
cbi-57	464	2	1	1	NUM
cbi-57	464	3	+	+	NUM
cbi-57	464	4	.	.	PUNCT
cbi-57	464	5	.	.	PUNCT
cbi-57	464	6	.	.	PUNCT
cbi-57	464	7	)	)	PUNCT
cbi-57	465	1	=	=	PUNCT
cbi-57	465	2	(	(	PUNCT
cbi-57	465	3	1	1	NUM
cbi-57	465	4	+	+	NUM
cbi-57	465	5	1	1	NUM
cbi-57	465	6	+	+	NUM
cbi-57	465	7	1	1	NUM
cbi-57	465	8	+	+	NUM
cbi-57	465	9	1	1	NUM
cbi-57	465	10	+	+	NUM
cbi-57	465	11	.	.	PUNCT
cbi-57	465	12	.	.	PUNCT
cbi-57	465	13	.	.	PUNCT
cbi-57	465	14	)	)	PUNCT
cbi-57	465	15	.	.	PUNCT
cbi-57	466	1	(	(	PUNCT
cbi-57	466	2	60	60	NUM
cbi-57	466	3	)	)	PUNCT
cbi-57	466	4	the	the	DET
cbi-57	466	5	value	value	NOUN
cbi-57	466	6	of	of	ADP
cbi-57	466	7	this	this	DET
cbi-57	466	8	series	series	NOUN
cbi-57	466	9	was	be	AUX
cbi-57	466	10	deduced	deduce	VERB
cbi-57	466	11	to	to	PART
cbi-57	466	12	be	be	AUX
cbi-57	466	13	!	!	PUNCT
cbi-57	467	1	in	in	ADP
cbi-57	467	2	(	(	PUNCT
cbi-57	467	3	3	3	NUM
cbi-57	467	4	)	)	PUNCT
cbi-57	467	5	.	.	PUNCT
cbi-57	468	1	likewise	likewise	ADV
cbi-57	468	2	,	,	PUNCT
cbi-57	468	3	if	if	SCONJ
cbi-57	468	4	the	the	DET
cbi-57	468	5	values	value	NOUN
cbi-57	468	6	for	for	ADP
cbi-57	468	7	each	each	DET
cbi-57	468	8	series	serie	NOUN
cbi-57	468	9	are	be	AUX
cbi-57	468	10	added	add	VERB
cbi-57	468	11	the	the	DET
cbi-57	468	12	result	result	NOUN
cbi-57	468	13	is	be	AUX
cbi-57	468	14	#	#	SYM
cbi-57	468	15	1	1	NUM
cbi-57	468	16	2	2	NUM
cbi-57	468	17	!	!	PUNCT
cbi-57	469	1	+	+	CCONJ
cbi-57	469	2	1	1	NUM
cbi-57	469	3	4	4	NUM
cbi-57	469	4	$	$	SYM
cbi-57	469	5	+	+	NUM
cbi-57	469	6	#	#	SYM
cbi-57	469	7	1	1	NUM
cbi-57	469	8	2	2	NUM
cbi-57	469	9	!	!	PUNCT
cbi-57	470	1	#	#	NOUN
cbi-57	470	2	1	1	NUM
cbi-57	470	3	4	4	NUM
cbi-57	470	4	$	$	NOUN
cbi-57	470	5	=	=	PUNCT
cbi-57	470	6	!	!	PUNCT
cbi-57	470	7	.	.	PUNCT
cbi-57	471	1	(	(	PUNCT
cbi-57	471	2	61	61	NUM
cbi-57	471	3	)	)	PUNCT
cbi-57	471	4	11	11	NUM
cbi-57	471	5	ongoing	ongoing	ADJ
cbi-57	471	6	and	and	CCONJ
cbi-57	471	7	future	future	ADJ
cbi-57	471	8	work	work	NOUN
cbi-57	471	9	11.1	11.1	NUM
cbi-57	471	10	proving	prove	VERB
cbi-57	471	11	#	#	SYM
cbi-57	471	12	1	1	NUM
cbi-57	471	13	"	"	PUNCT
cbi-57	471	14	=	=	SYM
cbi-57	471	15	0	0	PROPN
cbi-57	472	1	the	the	DET
cbi-57	472	2	first	first	ADJ
cbi-57	472	3	obvious	obvious	ADJ
cbi-57	472	4	point	point	NOUN
cbi-57	472	5	of	of	ADP
cbi-57	472	6	future	future	ADJ
cbi-57	472	7	work	work	NOUN
cbi-57	472	8	is	be	AUX
cbi-57	472	9	the	the	DET
cbi-57	472	10	proof	proof	NOUN
cbi-57	472	11	of	of	ADP
cbi-57	472	12	the	the	DET
cbi-57	472	13	conjecture	conjecture	NOUN
cbi-57	472	14	in	in	ADP
cbi-57	472	15	(	(	PUNCT
cbi-57	472	16	43	43	NUM
cbi-57	472	17	)	)	PUNCT
cbi-57	472	18	.	.	PUNCT
cbi-57	473	1	work	work	NOUN
cbi-57	473	2	on	on	ADP
cbi-57	473	3	this	this	DET
cbi-57	473	4	proof	proof	NOUN
cbi-57	473	5	is	be	AUX
cbi-57	473	6	ongoing	ongoing	ADJ
cbi-57	473	7	and	and	CCONJ
cbi-57	473	8	is	be	AUX
cbi-57	473	9	promising	promise	VERB
cbi-57	473	10	.	.	PUNCT
cbi-57	474	1	11.2	11.2	NUM
cbi-57	474	2	representing	represent	VERB
cbi-57	474	3	infinitesimal	infinitesimal	ADJ
cbi-57	474	4	values	value	NOUN
cbi-57	474	5	in	in	ADP
cbi-57	474	6	general	general	ADJ
cbi-57	474	7	,	,	PUNCT
cbi-57	474	8	the	the	DET
cbi-57	474	9	methods	method	NOUN
cbi-57	474	10	in	in	ADP
cbi-57	474	11	this	this	DET
cbi-57	474	12	paper	paper	NOUN
cbi-57	474	13	are	be	AUX
cbi-57	474	14	about	about	ADP
cbi-57	474	15	representing	represent	VERB
cbi-57	474	16	infinite	infinite	ADJ
cbi-57	474	17	values	value	NOUN
cbi-57	474	18	using	use	VERB
cbi-57	474	19	a	a	DET
cbi-57	474	20	series	series	NOUN
cbi-57	474	21	of	of	ADP
cbi-57	474	22	finite	finite	ADJ
cbi-57	474	23	terms	term	NOUN
cbi-57	474	24	.	.	PUNCT
cbi-57	475	1	however	however	ADV
cbi-57	475	2	,	,	PUNCT
cbi-57	475	3	it	it	PRON
cbi-57	475	4	volume	volume	VERB
cbi-57	475	5	2	2	NUM
cbi-57	475	6	,	,	PUNCT
cbi-57	475	7	issue	issue	NOUN
cbi-57	475	8	1	1	NUM
cbi-57	475	9	references	reference	NOUN
cbi-57	475	10	15	15	NUM
cbi-57	475	11	may	may	AUX
cbi-57	475	12	also	also	ADV
cbi-57	475	13	be	be	AUX
cbi-57	475	14	possible	possible	ADJ
cbi-57	475	15	to	to	PART
cbi-57	475	16	write	write	VERB
cbi-57	475	17	an	an	DET
cbi-57	475	18	infinitesimal	infinitesimal	ADJ
cbi-57	475	19	value	value	NOUN
cbi-57	475	20	in	in	ADP
cbi-57	475	21	a	a	DET
cbi-57	475	22	similar	similar	ADJ
cbi-57	475	23	way	way	NOUN
cbi-57	475	24	.	.	PUNCT
cbi-57	476	1	while	while	SCONJ
cbi-57	476	2	it	it	PRON
cbi-57	476	3	is	be	AUX
cbi-57	476	4	outside	outside	ADP
cbi-57	476	5	the	the	DET
cbi-57	476	6	scope	scope	NOUN
cbi-57	476	7	of	of	ADP
cbi-57	476	8	the	the	DET
cbi-57	476	9	present	present	ADJ
cbi-57	476	10	paper	paper	NOUN
cbi-57	476	11	,	,	PUNCT
cbi-57	476	12	there	there	PRON
cbi-57	476	13	is	be	VERB
cbi-57	476	14	some	some	DET
cbi-57	476	15	evidence	evidence	NOUN
cbi-57	476	16	that	that	SCONJ
cbi-57	476	17	,	,	PUNCT
cbi-57	476	18	for	for	ADP
cbi-57	476	19	instance	instance	NOUN
cbi-57	476	20	,	,	PUNCT
cbi-57	476	21	the	the	DET
cbi-57	476	22	series	series	NOUN
cbi-57	476	23	1	1	NUM
cbi-57	476	24	+	+	CCONJ
cbi-57	476	25	#	#	SYM
cbi-57	476	26	1	1	NUM
cbi-57	476	27	+	+	SYM
cbi-57	476	28	0	0	NUM
cbi-57	477	1	+	+	CCONJ
cbi-57	477	2	0	0	NUM
cbi-57	478	1	+	+	CCONJ
cbi-57	478	2	0	0	NUM
cbi-57	479	1	+	+	CCONJ
cbi-57	479	2	0	0	NUM
cbi-57	480	1	+	+	CCONJ
cbi-57	480	2	.	.	PUNCT
cbi-57	480	3	.	.	PUNCT
cbi-57	480	4	.	.	PUNCT
cbi-57	481	1	(	(	PUNCT
cbi-57	481	2	with	with	ADP
cbi-57	481	3	the	the	DET
cbi-57	481	4	zero	zero	NUM
cbi-57	481	5	repeating	repeat	VERB
cbi-57	481	6	forever	forever	ADV
cbi-57	481	7	)	)	PUNCT
cbi-57	481	8	represents	represent	VERB
cbi-57	481	9	an	an	DET
cbi-57	481	10	infinitesimal	infinitesimal	ADJ
cbi-57	481	11	value	value	NOUN
cbi-57	481	12	.	.	PUNCT
cbi-57	482	1	this	this	PRON
cbi-57	482	2	means	mean	VERB
cbi-57	482	3	that	that	SCONJ
cbi-57	482	4	the	the	DET
cbi-57	482	5	rules	rule	NOUN
cbi-57	482	6	established	establish	VERB
cbi-57	482	7	in	in	ADP
cbi-57	482	8	section	section	NOUN
cbi-57	482	9	8	8	NUM
cbi-57	482	10	are	be	AUX
cbi-57	482	11	limited	limit	VERB
cbi-57	482	12	to	to	ADP
cbi-57	482	13	cases	case	NOUN
cbi-57	482	14	where	where	SCONJ
cbi-57	482	15	the	the	DET
cbi-57	482	16	principal	principal	ADJ
cbi-57	482	17	value	value	NOUN
cbi-57	482	18	is	be	AUX
cbi-57	482	19	finite	finite	ADJ
cbi-57	482	20	or	or	CCONJ
cbi-57	482	21	infinite	infinite	ADJ
cbi-57	482	22	.	.	PUNCT
cbi-57	482	23	12	12	NUM
cbi-57	482	24	conclusion	conclusion	NOUN
cbi-57	482	25	here	here	ADV
cbi-57	482	26	a	a	DET
cbi-57	482	27	method	method	NOUN
cbi-57	482	28	of	of	ADP
cbi-57	482	29	summation	summation	NOUN
cbi-57	482	30	was	be	AUX
cbi-57	482	31	presented	present	VERB
cbi-57	482	32	that	that	PRON
cbi-57	482	33	uses	use	VERB
cbi-57	482	34	the	the	DET
cbi-57	482	35	structure	structure	NOUN
cbi-57	482	36	of	of	ADP
cbi-57	482	37	the	the	DET
cbi-57	482	38	hyperreal	hyperreal	ADJ
cbi-57	482	39	numbers	number	NOUN
cbi-57	482	40	to	to	PART
cbi-57	482	41	represent	represent	VERB
cbi-57	482	42	values	value	NOUN
cbi-57	482	43	for	for	ADP
cbi-57	482	44	divergent	divergent	ADJ
cbi-57	482	45	series	series	NOUN
cbi-57	482	46	.	.	PUNCT
cbi-57	483	1	this	this	DET
cbi-57	483	2	methodology	methodology	NOUN
cbi-57	483	3	was	be	AUX
cbi-57	483	4	shown	show	VERB
cbi-57	483	5	to	to	PART
cbi-57	483	6	be	be	AUX
cbi-57	483	7	stable	stable	ADJ
cbi-57	483	8	across	across	ADP
cbi-57	483	9	a	a	DET
cbi-57	483	10	variety	variety	NOUN
cbi-57	483	11	of	of	ADP
cbi-57	483	12	di!erent	di!erent	NOUN
cbi-57	483	13	scenarios	scenario	NOUN
cbi-57	483	14	.	.	PUNCT
cbi-57	484	1	one	one	NUM
cbi-57	484	2	unproven	unproven	ADJ
cbi-57	484	3	,	,	PUNCT
cbi-57	484	4	but	but	CCONJ
cbi-57	484	5	seemingly	seemingly	ADV
cbi-57	484	6	correct	correct	ADJ
cbi-57	484	7	,	,	PUNCT
cbi-57	484	8	conjecture	conjecture	NOUN
cbi-57	484	9	was	be	AUX
cbi-57	484	10	relied	rely	VERB
cbi-57	484	11	upon	upon	SCONJ
cbi-57	484	12	for	for	ADP
cbi-57	484	13	this	this	DET
cbi-57	484	14	formulation	formulation	NOUN
cbi-57	484	15	.	.	PUNCT
cbi-57	485	1	future	future	ADJ
cbi-57	485	2	work	work	NOUN
cbi-57	485	3	will	will	AUX
cbi-57	485	4	focus	focus	VERB
cbi-57	485	5	on	on	ADP
cbi-57	485	6	proving	prove	VERB
cbi-57	485	7	(	(	PUNCT
cbi-57	485	8	43	43	NUM
cbi-57	485	9	)	)	PUNCT
cbi-57	485	10	.	.	PUNCT
cbi-57	486	1	13	13	NUM
cbi-57	486	2	acknowledgements	acknowledgement	NOUN
cbi-57	486	3	i	i	PRON
cbi-57	486	4	wanted	want	VERB
cbi-57	486	5	to	to	PART
cbi-57	486	6	take	take	VERB
cbi-57	486	7	a	a	DET
cbi-57	486	8	moment	moment	NOUN
cbi-57	486	9	to	to	PART
cbi-57	486	10	thank	thank	VERB
cbi-57	486	11	stanley	stanley	PROPN
cbi-57	486	12	schmidt	schmidt	PROPN
cbi-57	486	13	.	.	PUNCT
cbi-57	487	1	i	i	PRON
cbi-57	487	2	was	be	AUX
cbi-57	487	3	thinking	think	VERB
cbi-57	487	4	on	on	ADP
cbi-57	487	5	this	this	DET
cbi-57	487	6	problem	problem	NOUN
cbi-57	487	7	at	at	ADP
cbi-57	487	8	the	the	DET
cbi-57	487	9	same	same	ADJ
cbi-57	487	10	time	time	NOUN
cbi-57	487	11	i	i	PRON
cbi-57	487	12	was	be	AUX
cbi-57	487	13	reading	read	VERB
cbi-57	487	14	his	his	PRON
cbi-57	487	15	life	life	NOUN
cbi-57	487	16	of	of	ADP
cbi-57	487	17	fred	fred	PROPN
cbi-57	487	18	books	book	NOUN
cbi-57	487	19	to	to	ADP
cbi-57	487	20	my	my	PRON
cbi-57	487	21	children	child	NOUN
cbi-57	487	22	.	.	PUNCT
cbi-57	488	1	the	the	DET
cbi-57	488	2	fundamental	fundamental	ADJ
cbi-57	488	3	idea	idea	NOUN
cbi-57	488	4	for	for	ADP
cbi-57	488	5	this	this	DET
cbi-57	488	6	method	method	NOUN
cbi-57	488	7	of	of	ADP
cbi-57	488	8	summation	summation	NOUN
cbi-57	488	9	came	come	VERB
cbi-57	488	10	from	from	ADP
cbi-57	488	11	thinking	think	VERB
cbi-57	488	12	about	about	ADP
cbi-57	488	13	gaastra	gaastra	PROPN
cbi-57	488	14	’s	’s	PART
cbi-57	488	15	original	original	ADJ
cbi-57	488	16	presentation	presentation	NOUN
cbi-57	488	17	(	(	PUNCT
cbi-57	488	18	gaastra	gaastra	PROPN
cbi-57	488	19	,	,	PUNCT
cbi-57	488	20	2016	2016	NUM
cbi-57	488	21	)	)	PUNCT
cbi-57	488	22	while	while	SCONJ
cbi-57	488	23	reading	read	VERB
cbi-57	488	24	life	life	NOUN
cbi-57	488	25	of	of	ADP
cbi-57	488	26	fred	fred	NOUN
cbi-57	488	27	:	:	PUNCT
cbi-57	488	28	kidneys	kidney	NOUN
cbi-57	488	29	to	to	ADP
cbi-57	488	30	my	my	PRON
cbi-57	488	31	children	child	NOUN
cbi-57	488	32	,	,	PUNCT
cbi-57	488	33	when	when	SCONJ
cbi-57	488	34	fred	fred	PROPN
cbi-57	488	35	was	be	AUX
cbi-57	488	36	using	use	VERB
cbi-57	488	37	the	the	DET
cbi-57	488	38	formula	formula	NOUN
cbi-57	488	39	for	for	ADP
cbi-57	488	40	arithmetic	arithmetic	ADJ
cbi-57	488	41	series	series	NOUN
cbi-57	488	42	(	(	PUNCT
cbi-57	488	43	schmidt	schmidt	PROPN
cbi-57	488	44	,	,	PUNCT
cbi-57	488	45	2012	2012	NUM
cbi-57	488	46	)	)	PUNCT
cbi-57	488	47	.	.	PUNCT
cbi-57	489	1	additionally	additionally	ADV
cbi-57	489	2	,	,	PUNCT
cbi-57	489	3	the	the	DET
cbi-57	489	4	infinite	infinite	NOUN
cbi-57	489	5	by	by	ADP
cbi-57	489	6	a.	a.	PROPN
cbi-57	489	7	w.	w.	PROPN
cbi-57	489	8	moore	moore	PROPN
cbi-57	489	9	provided	provide	VERB
cbi-57	489	10	some	some	DET
cbi-57	489	11	help	help	NOUN
cbi-57	489	12	to	to	ADP
cbi-57	489	13	the	the	DET
cbi-57	489	14	imagination	imagination	NOUN
cbi-57	489	15	in	in	ADP
cbi-57	489	16	his	his	PRON
cbi-57	489	17	discussion	discussion	NOUN
cbi-57	489	18	of	of	ADP
cbi-57	489	19	the	the	DET
cbi-57	489	20	löwenheimskolem	löwenheimskolem	NOUN
cbi-57	489	21	theorem	theorem	VERB
cbi-57	489	22	.	.	PUNCT
cbi-57	490	1	the	the	DET
cbi-57	490	2	basics	basic	NOUN
cbi-57	490	3	of	of	ADP
cbi-57	490	4	the	the	DET
cbi-57	490	5	discussion	discussion	NOUN
cbi-57	490	6	was	be	AUX
cbi-57	490	7	to	to	PART
cbi-57	490	8	point	point	VERB
cbi-57	490	9	out	out	ADP
cbi-57	490	10	that	that	SCONJ
cbi-57	490	11	there	there	PRON
cbi-57	490	12	was	be	VERB
cbi-57	490	13	little	little	ADJ
cbi-57	490	14	in	in	ADP
cbi-57	490	15	the	the	DET
cbi-57	490	16	theory	theory	NOUN
cbi-57	490	17	of	of	ADP
cbi-57	490	18	infinities	infinity	NOUN
cbi-57	490	19	that	that	PRON
cbi-57	490	20	were	be	AUX
cbi-57	490	21	really	really	ADV
cbi-57	490	22	unique	unique	ADJ
cbi-57	490	23	to	to	ADP
cbi-57	490	24	infinity	infinity	NOUN
cbi-57	490	25	.	.	PUNCT
cbi-57	491	1	even	even	ADV
cbi-57	491	2	finite	finite	ADJ
cbi-57	491	3	sets	set	NOUN
cbi-57	491	4	can	can	AUX
cbi-57	491	5	look	look	AUX
cbi-57	491	6	“	"	PUNCT
cbi-57	491	7	infinite	infinite	VERB
cbi-57	491	8	”	"	PUNCT
cbi-57	491	9	in	in	ADP
cbi-57	491	10	some	some	DET
cbi-57	491	11	ways	way	NOUN
cbi-57	491	12	to	to	ADP
cbi-57	491	13	other	other	ADJ
cbi-57	491	14	sets	set	NOUN
cbi-57	491	15	.	.	PUNCT
cbi-57	492	1	the	the	DET
cbi-57	492	2	techniques	technique	NOUN
cbi-57	492	3	and	and	CCONJ
cbi-57	492	4	ideas	idea	NOUN
cbi-57	492	5	explored	explore	VERB
cbi-57	492	6	in	in	ADP
cbi-57	492	7	section	section	NOUN
cbi-57	492	8	3	3	NUM
cbi-57	492	9	were	be	AUX
cbi-57	492	10	based	base	VERB
cbi-57	492	11	largely	largely	ADV
cbi-57	492	12	o	o	NOUN
cbi-57	492	13	!	!	PUNCT
cbi-57	493	1	of	of	ADP
cbi-57	493	2	thinking	think	VERB
cbi-57	493	3	about	about	ADP
cbi-57	493	4	infinities	infinity	NOUN
cbi-57	493	5	as	as	ADV
cbi-57	493	6	much	much	ADV
cbi-57	493	7	more	more	ADV
cbi-57	493	8	tame	tame	ADJ
cbi-57	493	9	and	and	CCONJ
cbi-57	493	10	finite	finite	ADJ
cbi-57	493	11	-	-	ADJ
cbi-57	493	12	like	like	ADJ
cbi-57	493	13	than	than	ADP
cbi-57	493	14	is	be	AUX
cbi-57	493	15	normally	normally	ADV
cbi-57	493	16	considered	consider	VERB
cbi-57	493	17	.	.	PUNCT
cbi-57	494	1	finally	finally	ADV
cbi-57	494	2	,	,	PUNCT
cbi-57	494	3	i	i	PRON
cbi-57	494	4	want	want	VERB
cbi-57	494	5	to	to	PART
cbi-57	494	6	thank	thank	VERB
cbi-57	494	7	jessica	jessica	PROPN
cbi-57	494	8	hastings	hastings	PROPN
cbi-57	494	9	,	,	PUNCT
cbi-57	494	10	whose	whose	DET
cbi-57	494	11	interest	interest	NOUN
cbi-57	494	12	in	in	ADP
cbi-57	494	13	the	the	DET
cbi-57	494	14	“	"	PUNCT
cbi-57	494	15	wheat	wheat	NOUN
cbi-57	494	16	and	and	CCONJ
cbi-57	494	17	chessboard	chessboard	NOUN
cbi-57	494	18	”	"	PUNCT
cbi-57	494	19	problem(weisstein	problem(weisstein	PROPN
cbi-57	494	20	,	,	PUNCT
cbi-57	494	21	2018	2018	NUM
cbi-57	494	22	)	)	PUNCT
cbi-57	494	23	originally	originally	ADV
cbi-57	494	24	introduced	introduce	VERB
cbi-57	494	25	me	i	PRON
cbi-57	494	26	to	to	ADP
cbi-57	494	27	the	the	DET
cbi-57	494	28	concepts	concept	NOUN
cbi-57	494	29	in	in	ADP
cbi-57	494	30	discrete	discrete	ADJ
cbi-57	494	31	calculus	calculus	NOUN
cbi-57	494	32	.	.	PUNCT
cbi-57	495	1	references	reference	NOUN
cbi-57	495	2	gaastra	gaastra	PROPN
cbi-57	495	3	,	,	PUNCT
cbi-57	495	4	l	l	PROPN
cbi-57	495	5	(	(	PUNCT
cbi-57	495	6	2016	2016	NUM
cbi-57	495	7	)	)	PUNCT
cbi-57	495	8	.	.	PUNCT
cbi-57	496	1	“	"	PUNCT
cbi-57	496	2	omega	omega	NOUN
cbi-57	496	3	:	:	PUNCT
cbi-57	496	4	how	how	SCONJ
cbi-57	496	5	hilbert	hilbert	PROPN
cbi-57	496	6	’s	’s	PART
cbi-57	496	7	infinite	infinite	ADJ
cbi-57	496	8	hotel	hotel	NOUN
cbi-57	496	9	can	can	AUX
cbi-57	496	10	be	be	AUX
cbi-57	496	11	used	use	VERB
cbi-57	496	12	to	to	PART
cbi-57	496	13	evaluate	evaluate	VERB
cbi-57	496	14	divergent	divergent	ADJ
cbi-57	496	15	series	serie	NOUN
cbi-57	496	16	”	"	PUNCT
cbi-57	496	17	.	.	PUNCT
cbi-57	497	1	in	in	ADP
cbi-57	497	2	:	:	PUNCT
cbi-57	497	3	grcc	grcc	PROPN
cbi-57	497	4	student	student	NOUN
cbi-57	497	5	mathematics	mathematic	NOUN
cbi-57	497	6	seminars	seminar	NOUN
cbi-57	497	7	.	.	PUNCT
cbi-57	498	1	url	url	PROPN
cbi-57	498	2	:	:	PUNCT
cbi-57	499	1	https://www	https://www	PROPN
cbi-57	499	2	.	.	PUNCT
cbi-57	499	3	youtube.com/watch?v=nlwh9oyiqye	youtube.com/watch?v=nlwh9oyiqye	PROPN
cbi-57	499	4	.	.	PUNCT
cbi-57	500	1	goldblatt	goldblatt	NOUN
cbi-57	500	2	,	,	PUNCT
cbi-57	500	3	r	r	NOUN
cbi-57	500	4	(	(	PUNCT
cbi-57	500	5	1998	1998	NUM
cbi-57	500	6	)	)	PUNCT
cbi-57	500	7	.	.	PUNCT
cbi-57	501	1	lectures	lecture	NOUN
cbi-57	501	2	on	on	ADP
cbi-57	501	3	the	the	DET
cbi-57	501	4	hyperreals	hyperreal	NOUN
cbi-57	501	5	:	:	PUNCT
cbi-57	501	6	an	an	DET
cbi-57	501	7	introduction	introduction	NOUN
cbi-57	501	8	to	to	ADP
cbi-57	501	9	nonstandard	nonstandard	ADJ
cbi-57	501	10	analysis	analysis	NOUN
cbi-57	501	11	.	.	PUNCT
cbi-57	502	1	springer	springer	NOUN
cbi-57	502	2	.	.	PUNCT
cbi-57	503	1	graham	graham	PROPN
cbi-57	503	2	,	,	PUNCT
cbi-57	503	3	r	r	NOUN
cbi-57	503	4	l	l	NOUN
cbi-57	503	5	,	,	PUNCT
cbi-57	503	6	d	d	PROPN
cbi-57	503	7	e	e	PROPN
cbi-57	503	8	knuth	knuth	PROPN
cbi-57	503	9	,	,	PUNCT
cbi-57	503	10	and	and	CCONJ
cbi-57	503	11	o	o	X
cbi-57	503	12	patashnik	patashnik	X
cbi-57	503	13	(	(	PUNCT
cbi-57	503	14	1994	1994	NUM
cbi-57	503	15	)	)	PUNCT
cbi-57	503	16	.	.	PUNCT
cbi-57	504	1	concrete	concrete	ADJ
cbi-57	504	2	mathematics	mathematic	NOUN
cbi-57	504	3	.	.	PUNCT
cbi-57	505	1	new	new	PROPN
cbi-57	505	2	york	york	PROPN
cbi-57	505	3	:	:	PUNCT
cbi-57	505	4	addison	addison	PROPN
cbi-57	505	5	-	-	PUNCT
cbi-57	505	6	wesley	wesley	PROPN
cbi-57	505	7	.	.	PUNCT
cbi-57	506	1	henle	henle	PROPN
cbi-57	506	2	,	,	PUNCT
cbi-57	506	3	j	j	PROPN
cbi-57	506	4	m	m	PROPN
cbi-57	506	5	and	and	CCONJ
cbi-57	506	6	e	e	PROPN
cbi-57	506	7	m	m	PROPN
cbi-57	506	8	kleinberg	kleinberg	PROPN
cbi-57	506	9	(	(	PUNCT
cbi-57	506	10	2003	2003	NUM
cbi-57	506	11	)	)	PUNCT
cbi-57	506	12	.	.	PUNCT
cbi-57	507	1	infinitesimal	infinitesimal	ADJ
cbi-57	507	2	calculus	calculus	NOUN
cbi-57	507	3	.	.	PUNCT
cbi-57	508	1	dover	dover	PROPN
cbi-57	508	2	publications	publication	NOUN
cbi-57	508	3	.	.	PUNCT
cbi-57	509	1	keisler	keisler	PROPN
cbi-57	509	2	,	,	PUNCT
cbi-57	509	3	j	j	PROPN
cbi-57	509	4	(	(	PUNCT
cbi-57	509	5	2012	2012	NUM
cbi-57	509	6	)	)	PUNCT
cbi-57	509	7	.	.	PUNCT
cbi-57	510	1	elementary	elementary	PROPN
cbi-57	510	2	calculus	calculus	PROPN
cbi-57	510	3	:	:	PUNCT
cbi-57	510	4	an	an	DET
cbi-57	510	5	infinitesimal	infinitesimal	ADJ
cbi-57	510	6	approach	approach	NOUN
cbi-57	510	7	.	.	PUNCT
cbi-57	511	1	second	second	ADJ
cbi-57	511	2	.	.	PUNCT
cbi-57	512	1	dover	dover	PROPN
cbi-57	512	2	books	book	NOUN
cbi-57	512	3	.	.	PUNCT
cbi-57	513	1	loeb	loeb	PROPN
cbi-57	513	2	,	,	PUNCT
cbi-57	513	3	p	p	PROPN
cbi-57	513	4	a	a	PROPN
cbi-57	513	5	and	and	CCONJ
cbi-57	513	6	m	m	PROPN
cbi-57	513	7	p	p	NOUN
cbi-57	513	8	h	h	NOUN
cbi-57	513	9	wol	wol	VERB
cbi-57	513	10	!	!	PUNCT
cbi-57	514	1	(	(	PUNCT
cbi-57	514	2	2015	2015	NUM
cbi-57	514	3	)	)	PUNCT
cbi-57	514	4	.	.	PUNCT
cbi-57	515	1	nonstandard	nonstandard	ADJ
cbi-57	515	2	analysis	analysis	NOUN
cbi-57	515	3	for	for	ADP
cbi-57	515	4	the	the	DET
cbi-57	515	5	working	work	VERB
cbi-57	515	6	mathematician	mathematician	NOUN
cbi-57	515	7	.	.	PUNCT
cbi-57	516	1	second	second	ADJ
cbi-57	516	2	.	.	PUNCT
cbi-57	517	1	london	london	PROPN
cbi-57	517	2	:	:	PUNCT
cbi-57	517	3	springer	springer	NOUN
cbi-57	517	4	.	.	PUNCT
cbi-57	518	1	paterson	paterson	PROPN
cbi-57	518	2	,	,	PUNCT
cbi-57	518	3	d	d	PROPN
cbi-57	518	4	a	a	PRON
cbi-57	518	5	(	(	PUNCT
cbi-57	518	6	2018a	2018a	NUM
cbi-57	518	7	)	)	PUNCT
cbi-57	518	8	.	.	PUNCT
cbi-57	519	1	“	"	PUNCT
cbi-57	519	2	banishing	banish	VERB
cbi-57	519	3	divergence	divergence	NOUN
cbi-57	519	4	part	part	NOUN
cbi-57	519	5	1	1	NUM
cbi-57	519	6	:	:	PUNCT
cbi-57	519	7	infinite	infinite	ADJ
cbi-57	519	8	numbers	number	NOUN
cbi-57	519	9	as	as	ADP
cbi-57	519	10	the	the	DET
cbi-57	519	11	limit	limit	NOUN
cbi-57	519	12	of	of	ADP
cbi-57	519	13	sequences	sequence	NOUN
cbi-57	519	14	of	of	ADP
cbi-57	519	15	real	real	ADJ
cbi-57	519	16	numbers	number	NOUN
cbi-57	519	17	”	"	PUNCT
cbi-57	519	18	.	.	PUNCT
cbi-57	520	1	in	in	ADP
cbi-57	520	2	:	:	PUNCT
cbi-57	520	3	arxiv	arxiv	PROPN
cbi-57	520	4	1108.5081v1	1108.5081v1	NUM
cbi-57	520	5	.	.	PROPN
cbi-57	520	6	url	url	PROPN
cbi-57	520	7	:	:	PUNCT
cbi-57	520	8	https://arxiv.org/abs/	https://arxiv.org/abs/	NOUN
cbi-57	520	9	1108.5081	1108.5081	NUM
cbi-57	520	10	.	.	PUNCT
cbi-57	520	11	paterson	paterson	PROPN
cbi-57	520	12	,	,	PUNCT
cbi-57	520	13	d	d	PROPN
cbi-57	520	14	a	a	X
cbi-57	520	15	(	(	PUNCT
cbi-57	520	16	2018b	2018b	NUM
cbi-57	520	17	)	)	PUNCT
cbi-57	520	18	.	.	PUNCT
cbi-57	521	1	“	"	PUNCT
cbi-57	521	2	banishing	banish	VERB
cbi-57	521	3	divergence	divergence	NOUN
cbi-57	521	4	part	part	NOUN
cbi-57	521	5	2	2	NUM
cbi-57	521	6	:	:	PUNCT
cbi-57	521	7	limits	limit	NOUN
cbi-57	521	8	of	of	ADP
cbi-57	521	9	oscillatory	oscillatory	ADJ
cbi-57	521	10	sequences	sequence	NOUN
cbi-57	521	11	and	and	CCONJ
cbi-57	521	12	applications	application	NOUN
cbi-57	521	13	”	"	PUNCT
cbi-57	521	14	.	.	PUNCT
cbi-57	522	1	in	in	ADP
cbi-57	522	2	:	:	PUNCT
cbi-57	522	3	arxiv	arxiv	PROPN
cbi-57	522	4	1108.4952v1	1108.4952v1	NUM
cbi-57	522	5	.	.	PUNCT
cbi-57	522	6	url	url	PROPN
cbi-57	522	7	:	:	PUNCT
cbi-57	522	8	https://arxiv.org/abs/	https://arxiv.org/abs/	NOUN
cbi-57	522	9	1108.4952	1108.4952	NUM
cbi-57	522	10	.	.	PUNCT
cbi-57	523	1	schmidt	schmidt	PROPN
cbi-57	523	2	,	,	PUNCT
cbi-57	523	3	s	s	X
cbi-57	523	4	(	(	PUNCT
cbi-57	523	5	2012	2012	NUM
cbi-57	523	6	)	)	PUNCT
cbi-57	523	7	.	.	PUNCT
cbi-57	524	1	life	life	NOUN
cbi-57	524	2	of	of	ADP
cbi-57	524	3	fred	fred	NOUN
cbi-57	524	4	:	:	PUNCT
cbi-57	524	5	kidneys	kidney	NOUN
cbi-57	524	6	.	.	PUNCT
cbi-57	525	1	polka	polka	NOUN
cbi-57	525	2	dot	dot	NOUN
cbi-57	525	3	publishers	publisher	NOUN
cbi-57	525	4	.	.	PUNCT
cbi-57	526	1	weisstein	weisstein	ADV
cbi-57	526	2	,	,	PUNCT
cbi-57	526	3	e	e	PROPN
cbi-57	526	4	w	w	PROPN
cbi-57	526	5	(	(	PUNCT
cbi-57	526	6	2018	2018	NUM
cbi-57	526	7	)	)	PUNCT
cbi-57	526	8	.	.	PUNCT
cbi-57	527	1	“	"	PUNCT
cbi-57	527	2	wheat	wheat	NOUN
cbi-57	527	3	and	and	CCONJ
cbi-57	527	4	chessboard	chessboard	ADJ
cbi-57	527	5	problem	problem	NOUN
cbi-57	527	6	”	"	PUNCT
cbi-57	527	7	.	.	PUNCT
cbi-57	528	1	in	in	ADP
cbi-57	528	2	:	:	PUNCT
cbi-57	528	3	mathworld	mathworld	NOUN
cbi-57	528	4	—	—	PUNCT
cbi-57	528	5	a	a	DET
cbi-57	528	6	wolfram	wolfram	PROPN
cbi-57	528	7	web	web	NOUN
cbi-57	528	8	resource	resource	NOUN
cbi-57	528	9	.	.	PUNCT