C ubo Matemática Edu cacional 
Vol. 5., N° 1, Enero 2003 

A N e w Expansion Formula 

G e orge A . Anastassiou 
Dcvm·tmenl o/ Mathematu;al S cien ces 

Uniuc 1·sby o/ M emphis 

Mcmvhis, TN 38 152 U.S. rt. 
c-ma·íl: ganaslss@mcmphis. ed1J 

2000 AMS S ubject C lassilication : 26A42, 26A99, 26D 15 

Key Words and Phr ases: T aylo r li ke expan ion , Rie ma nn- SLieltjes integral , 
inequaliLy íor integ ral ancl S\!Im. 

A bstract 

A new Tay lor like exp1:1insion íormula is est.ablished . Rere t!he JUemann-
St.ieltjes i.ntegral of a funcllion is expanded into a finit.e sum for m wh ich 
involves the deriva.tives ar llhe funct ion evaluated al t.he rigbt end point of 
t he int.erwl of integrnllimi. Thc error of t hc appro.ximation is given in un 
integral fo rm involv ing tlH~ nllh deriva.ti ve of thc íunction. lmp!icullions and 
applicntio ns o f tbe fo r mu la follmv. 

l. R esults 

We give our first end mai n resulL: 

T beorem l. Let go be a Lebesgue integrable and of bou.nded variation function 
on [a, bJ, a < b. We fonn 

g¡ (X) :=f.' go(t.)dt,. ( 1) 
j ' (x - t)•- 1 g.(x) := 

0 
--¡;;-::-¡y-oo(l)dl, n EN, x E [a, b[ . (2) 



26 George A. Anastassiou 

Let f be such that ¡Cn- I) is a absolutely continuous function on [a, bj. Then 

[ fdgo = ~(- !)• ¡<•l( b)gk(b) - f(a)g0(a) 
k:::O 

+ (-1)" l 9n-1(t)J(n)(t)dt. 
Proof. We apply integration by parts repeatedly (see [lj, p. 195): 

t fdgo = f(b)go(b) - /(a)go(a) - l gof'dt, 
and 

1• gof'dt 

Furthermore 

1· J"g¡dt 

So far we have got 

1• J'dg1 = J'(b)g1(b) - /'(a)g1(a) - [ 91/"dt 
¡'(b)g1(b) - t 91/"dt. 
1• J"d92 = J"(b)92 (b) - /"(a)g2(a) -1• 92/"'dt 
J"(b)92(b) -1· g2J'"dt. 

t fdgo f(b)9o(b) - f( a)go(a) - J'(b)g1(b) 
+ !"(b)92(b) -1· 92/"'dt. 

Similarly we find 

J.' 92/"'dt = ¡· f"'dg, = f"' (b)g3(b) -1· g,¡C4ldt. 
T hat is, 

t fd!Jo = f(b)go(b) - /(a)g0 (a) - ¡'(b)91(b) 
+ /"(b)92(b) - J"'(b)g3(b) + t g,¡C4ldt. 

(3) 



A Ncw Expnnsion Fbrmu/a 27 

'fhc validit.y of (3) is now cleer. o 
On Theorem 1 wc have 

Coro llnry l. Additionally a.ssurne that ¡C11> exists and rs boun ded. 1'lien 

ll fdYo - ~(- 1 )•¡<kl(b)9•(b) + / (a )90(a)I 
ll 9,. - 1(t)¡!")(l)dl,¡ S ll f'ºlll t l9n - 1(t)ldL (4) 

As a continuation of Corollary 1 wc have 

Corollnry 2. Assuming Otal go is bou.nded we obtain 

lt /dgo - ~(-1)" ¡ lkl (b)g.(b) f / (a)90(a) I 
S 11/(")lloo ll go ll oo (b - ,a)" , 

'" 
Proof. llerc we est.i matc the ri ght-hand side of incquality {•l ). 

Wc see that 

That i 

Conscqucnt ly, 

l9n- 1(x)I 11.z (x - L)"- 2 90(t)dt l 
0 (n-2)! 

S [ (~n-~)2~~2 lgo(t)ldt 
S llgo ll oo J.' (x _ t)" -'dt 

(n - 2)! • 

~(x - a)"- 1 , 
(n - 1)! 

J." 19,,_ , (t)ldt S ll go ll (b ~!a)" , 

J\ rcfinemcnt of (5) follows in 

(5) 

o 



28 George A. Anastassiou 

Cor o llary 3. Here tite assumptions are as in Corollary 2. We additionaily 
assume that 

i.e.1 f possess infinitely many derivatives and alt are uniformiy bounded. The1i 

11• fdgo - ~(-1)• ¡(•)(b)gk(b) + f(a)g.(a)/ 
(b - a)" 

~ Kllgo lJoo-n_!_ ' \In::". l. 

A consequence of (6) comes next. 

(6) 

Corollary 4 . Same assumpt1:ons as in Corollary 3. For sorne O < r < 1 it holds 
that 

n (b - a )" 
O(r ) = KIJgoll 00 - n- !- ~O, a.s n ~ +oo. (7) 

That is 
ri-1 b 

lim "'(- 1)• ¡(k)(b)gk(b) =J. fdgo + f(a)go(a). (8) 
n--+oo L-

k = O ª 

Proof. Call A := b - a > 01 we want t o prove that ~ --+ O as n--+ +oo. Set 

See that 

A" 
Xn := nf' n = 1,2, . 

Xn+ 1 = ___.:'.'.!__·Xn , n = l,2,. 
n+l 

But there exists no EN: n > A - 1, i.e., A< 1l{) + 1, that is , r := n:+l < 1 (in 
fac t Lake no := rA - Il + 1, where r·l is t he ceiJj ng of the number) . Thus 

Xno+ l = re, 

where 

C := X,..0 > 0. 
T herefore 



A No w Expansion fbrmula 29 

Likcwi we gct 

And in g ncral wc obtain 

O < X,.0 ¡ k < rk · e e· · r 010 tk, k E N 

whcrc e· : *°· That is 
inccrN-oasN - · ¡ , wcobtainthatx,.,.-OasN- l . l .c.,~ -o, 
~n-1 O 
Re mark l. (O n Thooro1n 1) t;u rthcrmorc w 1ha1 (hcrc f E C"(la,bl)) 

lt fd!JO 1 /(a)9o(a) I 
c3> lf:( - 1J' ¡c'><bJg,(b) 1 (-1)" l g._,(1J1MuJiu l 

l.· o o 

:S ~11/(k)ll l9•(b)l 1 ll / 1"lll f 19n - 1(l.)jdl. 
k o • 

Call 
L : mox{ll/ 11 , llf'll ,. ... 11 / 1">¡¡ ). (9) 

Thcn 

11' fdgo 1 / (a)9o(a)I 
:S /, . { ~¡9,(b)l I f 19n -1(l) jd1} . n El'I fi xcd. (10) 

2. Applications 

(I} Lcl {ttn.}nieN be a scquo11cc of 13orcl finitcsigncd mcasu rcs on [a, bl. Consiclcr 
thc dislribution fun tions g0 ,,,.(x) : ''"'¡a,xl, x E a, bJi m. EN, which 1:1rc of 
boundl'<I ''"riolion and Lebcsgue integrable. Clcarly ( hcre f E C" !a, bl) 

1' fd¡1,,, t fdg.,.,, 1 g0 ,,,,(a) · / (a). (11 ) 



30 George A. A nllStBSSiou 

We wo uld like to study the weak corwergence of µ,,.,. t.o zero, as m -+ +oo. F'rom 
(JO) a nd (11) we get 

lt fdl'm l $ L · {~ 19<,m(b) I + t l9n-l ,m(t)ldt} , 'Vm EN. (1 2) 
Here we assume t. hat 

and 

Let k E N, then 

by (2) . 
Thus 

That is1 

Next we have 

Hence 

9o,m(b) - O, 

J.' l9o,m('t) ldt - O, as m - +oo. 

( (' )(b-a¡•-
1 

l9k,m(b) I $ l o l9o,m(t) ldt T-Ji! - O. 

19•,m(b)I - @, 'Vk EN, m - +oo. 

r (x -t)"-2 
9n- 1,m(X) = la (n _ 2)! 9a,m(t)dt . 

r (x - t)"-2 
l9n-l,m(X)I $ l o (n _ 2)! l9o,m(t)ldt 

(x - a)"- 2 ( ' 
$ ~la l9o,m(t)ldt 

(b - a¡n-2 j' 
$ --¡n=2j'! 

0 
l9o,m(t)ldt . 

Consequently we get 

l b (b - a¡•- 1 J.' 
0 

19n-1,m(X)ldx $ (n- 2)! a l9o,m(t)ldt - O, 

asm-+ . I.e. , 

J.' l9n-1,m(t)ldt - O, as m - +oo. 

(13) 

(14 ) 

( 15) 



A New Exf)8nsion Fbnnula 

Pinally from ( 12) , (13), ( 1'1) and (15) we d erive that 

i.' fdµ,,, - O, as m -
ThaL is, µm converges weakly Lo zero as m - + . 

T he lasL r ulL was firsL proved (case of n = O) in l2J a nd Lhen in l3J. 

31 

( 11 ) Pormula (3) is expecLed LO huve npplications tio Nume rical l nLegration. 
( 111 ) Lec la, bJ = JO, l J a nd go(t) = t . Lct f be uch LhaL ¡ l"- 1> is a absoluwly 

co ntinuous íuncLion on [0 1 q. Then by Thoorem J we find 9u{X) = ~, ali 
n E N. Fu rLhermore we geL 

1 ,,_ , ¡ i• l( ) ( )" 1' f fdt = l::Hl•--1- + ...=.!__ t." ! 1" >(t )dt. 
lo k = O ( k + 1 )! n! O 

( 16) 

One can d eri ve o t her formulas like ( 16} ror various basic ga 1s. 

References 

111 Apostol, T. M .1 11Mathemat1cai AuaJy 1s", Ad d ison- Wesley, Rea.ding, M A1 
1960. 

(21 Hogoas, G. 1 11CharacterizaL'ion o/ weak conuergen.ce of si91ied measures on 
JO, q•, Math. Sean<!. 41. 175- 184, 1977. 

PI Joboson, J., An elemer1tar·y charocten.zal ion o/ weak con:uerycmce o/ mea-
s ures, The Ame rican Mat.h. Mont hly, 92 , :--.10 2 , 136- 1371 !985.