Increasing the convergence rate of series

Increasing the Convergence Rate of Series I. M. Longman Raymond and Beverly Sacklm Faculty of Exact Sciences Department of Geophysics G Planetary !Science.s Tel Aviv University Tel Aviv, 69978, Israel

ABSTRACT This paper deals with the question of obtaining from the sequence (s, } of partial sums of a convergent series s a new sequence ( t,, } which converges to the same limit s as s, , but more rapidly. When the general term u, of the series s possesses certain types of expansion involving inverse powers of n, it is shown how t,, is obtained by adding a fixed number of terms to s,. When the series s is convergent, these terms tend to zero as n tends to infinity, but they are such as to make t,, much more rapidly convergent to s-in fact we can make the convergence rate as great as we wish. Explicit general formulas are obtained for a wide range of important series.

1.

INTRODUCTION

In previous papers [2-41, the author has introduced methods for the summation and also convergence acceleration for a wide range of series. However, as pointed out in those papers, many cases, notably monotonic series, are not amenable to the above type of convergence acceleration, while quite often the above mentioned summation method runs into severe practical djfficulties when certain inverse Laplace or Mellin transforms are not readily available in explicit closed form. For details the reader is referred to the papers cited above. The present paper sets out an entirely different method of convergence acceleration which is, in particular, very suitable for monotonic series, but is not restricted exclusively to them. Suppose we wish to sum a slowly convergent series OCI

s=

CUk

0)

k=l

APPLIED

MATHEMATICS

AND COMPUTATION 24:77-89

0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017

(1987)

77 OOQfKW3/87/$03.50

78

I. M. LONGMAN

having partial sums

n =1,2,3

,...

.

(2)

Our method is to add to s, an appropriate quantity to obtain t,, and to consider, instead of the sequence { s, }, the modified sequence ( t, }. In general we seek to arrange matters so that t, also tends to s as n tends to infinity, but faster than s, does. We maximize the rate of convergence of t, in a certain sense to be explained below. Some treatment will also be given, by way of examples, for the “summation” of divergent series, but then of course the addition that we make to { s, } to obtain the convergent sequence { t, } will not tend to zero as n tends to infinity. Specifically, let us start with a series (1) in which u,+ 1 possesses the expansion u n+l=

f

bindawi,

i=l

(3)

where (Y is not necessarily an integer, and (Y and the bi are supposed to be known or readily obtainable. Then for t, we assume an expression K t, =

C

s, +

uinn-n-i,

i=O

(4

where K is some positive integer. Assuming for the moment that (Y is positive, it is clear that t, will tend to s as n tends to infinity, but we are at liberty to choose the a, so as to maximize the convergence rate of t, in some sense. Let us consider the difference

t n+l

-t,

=%a+1 +

5 ai[(n+l)-aei-n-‘-i] i==O

= E binem-‘+ i-l

f

ain-‘-i[(l+n-‘)~n~‘--l].

(5)

i=O

Since t, tends to s, tn+l - t, must tend to zero, and if we choose the a, to

Increasing the Convergence Rate of Series

79

make t. + 1 - t. as close to zero as we can, we maximize in this sense the rate of convergence of t. to s. Now

(l+n-,)-"-'_l= k~= 1 (-a-i) k

(e)

n-k'

where (-a-i)k

---( - 1 ) k ( a + i ) ( a + i + l ) ' ' ' ( a + i + k - 1 ) k !

(7)

so that

b,n-"-' + E a,n _ a _

t.+,-t.= E i~l

i~O

i

--

n-k.

(8)

k=l"

The coefficient of n - a - j ill (8) is

'-'(

bi+ i E a, ~O

--a--i

j-i

)

(9)

'

where we now assume that I~j~
(10)

We now demand that the expression (9) be zero for j = 1,2 ..... K + 1. This gives rise to a triangular system of equations which can be solved successively to obtain a~. We note, in passing, that due to the triangular nature of the system, the values obtained for the a~ are independent of K, as well as being easily calculable. In fact we will find an explicit expression for the a i in terms of the b j, a, After some careful analysis we find that i

a'=(a+i)-i

( a +ri ) b ' - ' + l ' ~-" ( - I ~" ' B"~

i = 0 , 1 ..... K.

(11)

r~0

Here B, are the Bernoulli numbers: Bo = 1, BI-2,-! Bs--6!, B4 = -3o,± Be = ~42, B8 --_ - ~1, Bw = ~ , BI 2 = _ 2730' ~1 etc. It should be noted that B2.+1 =

80

I.M. LONGMAN

0 for n = 1, 2, 3 . . . . . An extensive tabulation of the Bernoulli numbers is given in Abramowitz and Stegtm [1]. Since the b i are assumed known, (11) gives us the a i for use in (4), and we can use as many a i as we wish, depending on the rate of convergence we wish to achieve. It should be borne in mind that the use of (4) together with (11) is completely different from the Euler-Maclaurin formula, which requires the evaluation of an integral, and which gives rise usually only to an asymptotic series. It should likewise be noted that a fixed finite value of K should be used, and that (4) with fixed n and infinite K is likely not to converge. A number of examples of use of the above method will be given in the next section.

Turning to the summation of power series, let us now assume that un+ l in (1) can be expanded in the form

u.+ 1

bin

.

(12)

t. = s. + x" ~ ain -~-i.

(13)

~n L i=0

Then proceeding as before, let K i=0

Then K

t.+l-t.=u.+

x+x" E ain-~-i[x(l+n-l)-"-i-1]

•

(14)

i=O

Expanding the binomial as before, we find

x"

=

bin-~-i + ~-" a i n - " - i x i =o

k

n - k + x - 1 . (15)

a i - a j,

(16)

iffi0

1

Here the coefficient of n - " - j is, for 0 ~< j ~< K,

bi+

'(

x~,

iffi0

-a-i j-i

)

Increasing the Convergence Rate of Series

81

and equating this to zero for ] = 0 , 1 , 2 ..... K yields once again a triangular system of equations which can be solved successively for the ai. The first few results are bo a 0 -

1--x'

b1

a2~

~xbo

1- x

(1 - x) ~'

b2

(a + 1)xb I

1-x

11(1- x) ~

- -

a(a+l)

the general solution being

ai=(1-x)-

1 ~ (_l),(a+i_l~ ,ffi0

A,(x) b

r

/(l-x)'

i = 0 , 1 , 2 ..... K.

~-"

(17) Here the A,(x) are polynomials of degree r in x, the first few being

Ao(X ) = 1, Adx)

A z ( x ) = x +4xZ + x 3,

= x,

A2(x ) = x+ x 2,

A4(x ) = x + l l x

~+11x 3 + x 4.

Ai(x ) is the coefficient of (-1)'(a+i-1)(1-x)-i-lb

o

in a~, and we readily find the relation

a,(x)

,

xZ k-I

x)

ak_dx),

1,2,3 .....

(18)

k

Equation (18) is not very convenient for calculating successive At(x), but the

82

I.M. LONGMAN

following recurrence relation holds: Ar+I(X)=xA~(x)+xr+IA'(x-1),

r = 1,2,3 .....

(19)

or, equivalently, ff we write

At(x) = E a,kxk,

r > 0,

(20)

k=l

then a t + l , k = kark + ( r -- k + 2 ) a r , r_k+ 2

(21)

for k = 1,2 . . . . . r + 1. Here any ars for s > r is taken to be zero. An important particular case arises when x = - 1 , as this is often appropriate for alternating series. As a check on the coefficients in At(x) for any given r we can use the fact, which follows immediately from (18), that

Ar(1)=r!.

(22)

Table 1 gives the coefficients ark of Ar(x ) for r = 1 through 10. Each row is derivable from the previous row using (21). As a check Erk=lark = r! has been verified for each row, while the last column of the table gives

ark ( - 1) k = A,( - 1),

(23)

k=l

for use in alternating series. The coefficients satisfy the symmetry relation ark = ar,r_k+ 1

which follows from (21) on replacing k by r - k + 2. For this reason only coefficients for 1 ~< k < 5 are shown. In the next section some specific examples are given, and an interesting application is given to divergent series associated with the Riemann zeta function. This application brings out a connection between the Bernoulli numbers and the Riemann zeta functions of negative integral orders.

83

Increasing the Convergence Rate o f Series TABLE 1 CoErrtcw.rrrs ark o r A,(x) r o a r = 1(1)10 a ark

~ r

k=l

1

1

2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 1 1

2

3

4

5

(

-

1)kark

k-I -1

1 4 11 26 57 120 247 502 1013

1 11 66 302 1191 4293 14608 47840

1 26 302 2416 15619 88234 455192

1 57 1191 15619 156190 1310354

0 2 0 - 16 0 272 0 - 7936 0

aSee Equation (20).

2.

NUMERICAL EXAMPLES W e start w i t h the series o0

s = ~

~2

n -~ = - - = 1.0449340~7,

n=l

(24)

6

whose c o n v e r g e n c e is rather too slow for convenient direct computation. I n o r d e r to simplify matters, it is convenient to take un+ 1 = n -u, rather t h a n ( n + 1) -2. This simply amounts to starting our series at u 2 a n d writing n-1

s.=

E

k -~,

(~5)

k=l

a n d makes no essential difference to our results. This t y p e of simple modification is often usehtl. W e take a = 1, b I = 1, b i = 0, i > 1. T h e n according to (11)

a, = ( - 1 ) ' B , ,

(~o)

84

I.M. LONGMAN TABLE 2 APPROXIMAr,rrsTO ~'(2) = ~r2/6 n

Sn

t.

2 3 4 5 6 7

1.000 1.250 1.361 1.424 1.464 1.491

1.644912574 1.644933747 1.644934052 1.644934065 1.644934066 1.644934067

Bo B1

B2 B4 B6

so that

t,=s,+

n

n2 t-~-S ~ ~7 - ++

+ .--,

(27)

where only a finite number of terms is added to s , to obtain t,. Adding, for example, six terms to s n, we have n

1

t,= ~.~ k-2+n-X+~n-2+~n

-~nl -5+~n-7_~nl

-9,

(28)

k=l

and Table 2 is obtained. A m u c h more dramatic example is provided by the extremely slowly convergent series

s= ~

n - H = ~(1.1) = 10.58444846.

(29)

n=l

O n c e again it is convenient to take the sum of the first n - 1 terms as s,, and write u , + 1 n - 1 . 1 , so that there is no u r Here we take in (3) a = 0.1, b 1 = 1, b i = 0, i > 1. Choosing K = 8, we find from (11) n-1

t.= E i=1

8

i-H+

Z ai n - ( ' + ° a ) i=0

(30)

85

Increasing the Convergence Rate of Series TABLE 3 xPpx~oxtMAacrs TO ~'(1.1) = 10.58~a.A.846 vsmc K = 8 n

s,

t,

2 3 4

1.000 1.467 1.765

10.58444283 10.58a,~835 10.58a~aA846

with a o = 10, a l = 0.5,

a5 = O, 1.1×2.1×3.1×4.1×5.1

1.1 a2=

a6 =

12 '

30,240

a 7 = 0,

a 3 = 0,

1.1 × 2 . 1 x 3 . 1 a4=

720

--1.1×2.1× '

as =

."

×7.1

1,209,600

'

and Table 3 is obtained. T h e following example demonstrates a convenient way to calculate Euler's constant

y=

n

~"m

1

1

1

1 + 2- + 3- +''" + - - I n n n

) =0.577215665.

(31)

It is slightly more convenient to consider the equivalent [ 1 1 •y=,,~_.m 1 + ~ + ~ + . . .

1 +---ln(n+ln

)] '

(32)

from which we see that ~, is the sum of the slowly convergent series s = (1- t,2)+ (~- ~)

+ ( ~ - ~,~) + • • •.

(33)

(l+ n-'),

(34)

W e m a y take

n-l-t

86

I. M. LONGMAN

which has the expansion

Un+ 1

in-2

1 -3 1 -4 -- ~n -~- ~/~

(35)

....

T h u s we take

a=l,

b~

( - 1 ) ~+l i+1 '

i=1,2,3 .....

(36)

a n d from (11) we find for K = 3

(1

t,=(1-1n2)+ 1

+ in

-1

~-ln

+

1/1-2__

(1

+'"+ ~

12on

-4

-n --1

-- In -

n)

-

n--1

(37)

,

from which tl5 = 0.577215665. Of course, by taking a larger value of K the c o n v e r g e n c e rate can be increased considerably. It is instructive now to consider a simple divergent series, viz. s=1+2+3+

.-. +k+-..,

(38)

for which it is c o n v e n i e n t to take u . + x = n and a = - 2. Of course this value of a is outside the range required for t. - s . to tend to zero as n tends to infinity, b u t this is not surprising, since now our series is divergent a n d s . t e n d s to infinity. Now we have b I = 1, bi = 0, i > 1, a n d so from (11) we obtain 1

1

a o = _ i B o = _ ~, at=

D1 -BI-~, I

1

a 2= _TB 2=-~q,

while a i = 0 for i > 2. T h e existence of only a finite n u m b e r of nonzero a i is of course the situation w h e n e v e r a is a negative integer, a n d this means that if we take K sufficiently large, t . is constant a n d represents the exact "'sum" of our series in this sense. To obtain a finite " s u m " in this way we need to have b~ = 0 for

Increasing the Convergence Rate of Series

87

all i > - a. In the present case we have for K >1 2 n-I

t. = E k - i n ~ + ~,~ k=l

=~n(n- 1 ) -

~n2+ ~ n - ~ =

'

(30)

and this is, in a certain sense, the " s u m " of one series (38), since ~( - 1) = - __t 12" More generally for the series oo

s=

E ,,"

(4o)

for r a positive integer, we could expect to have the " s u m " ~( - r). In fact, if we follow through our procedure with u , + 1 = n', a = - r - 1, b I = 1, b~ = 0, i > 1, we find from (11)

at+l=

( - 1)'Br+ 1 r +1

(41)

for r odd, while

a.+~=o

(42)

for r even. Thus for K >i r + 1 we will find ( - 1)'/~,+ ~

t. =

(~)

r+l

for all positive integral r. Thus we expect to have

( - 1)" B ~(-r)---

r+l

,+i,

= 1 , 2 , 3 .... ,

(44)

and this is correct [1]. T h e following example is chosen to show how an alternating series whose terms decrease steadily in absolute value can be treated in two distinct ways b y the methods of this paper. (Of course, there are other ways for summing

88

I.M. LONGMAN

a l t e r n a t i n g series. See for example L o n g m a n [3].) T h e first w a y is to c o m b i n e successive t e r m s in order to obtain an equivalent monotonic series, while the s e c o n d w a y consists in the use of (13), (17) for x = - 1. T h e series s = 1 - ~ + ~ - ¼+ . . . .

(45)

l n 2 = 0.693147181

c a n b e w r i t t e n in the form oi a monotonic series 1 s=

1 ~

1 + .--~7~ + . . - .

3×4

1×2

(46)

oxIJ

H e r e it is c o n v e n i e n t to take

1

1

=

2n(2n

4n 2

1)

1

+

1

+

+.-.

(47)

07~

w i t h no u 1. Thus w e have here a = 1, b 1 = ¼, b 2 = ~, b 3 = t , etc., a n d w e easily find from (11) _l

0-0--

~ ,

al

3_

~

16,

i

a2--

9

8 ,

a 3 ~--- i ~

~

a4~---

3~

.

U s i n g these a i, w e have for K = 4 1 t. = s. + --

4n

3 +

1

9

+ - -

~

1 + - -

8n 3 + ~

32n 5'

(48)

where n-I

1

Z

(2k-1)2k"

(49)

k=l

T h e n w e find t m = 0.693147180. Of course, quicker convergence is achieved if w e take a larger value of K. T o a p p l y t h e second m e t h o d to (45) w e take in (12)

-(-1) ° x =

-

I,

u.+~

=

, 11

~ = 1,

(50)

Increasing the Convergence Rate of Series

89

so t h a t b o = - 1, b i = 0, i > 0. Choosing K = 4, w e find from (17) a o = - ~, a l = - ¼ , a a = 0 , a 3 = ~, a 4 = 0 , so that

1 1 2n

4n 2 +

(51)

where

1 s,=l-~+3

i

( - 1)" " "+

n-1

'

n=2,3,4

.....

(52)

W e readily find that t ~ = 0.693147181, a n d o n c e again the convergence rate would b e further i m p r o v e d b y using m o r e a iREFERENCES M. Abramowitz and I. A. Stegun, Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1968. 2 I . M . Longrnan, The summation of power series and Fourier series, 1. Comput. Appl. Math. 12&13:447-457 (1985). 3 I. M. Longman, The summation of series, Appl. Numer. Math., 2:135-141 1

4

(1986). I . M . Long/nan, The summation of Fourier, Chebyshev and Legendre series, Appl. Math. & Comp., to appear.