Johnson Baugh

Summing an Alternating SeriesAuthor(s): Richard JohnsonbaughSource: The American Mathematical Monthly, Vol. 86, No. 8 (Oct., 1979), pp. 637-648Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2321292Accessed: 11/08/2010 01:21

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1979] SUMMING AN ALTERNATING SERIES 637

2. G. H. Hardy, J. E. Littlewood, and G. P6lya, Inequalities, Cambridge University Press, New York, 1952. 3. Felix Hausdorff, Summationsmethoden und Momentfolgen, I, Mathematische Zeitschrift, 9 (1921) 74-109. 4. J. W. Pratt and J. S. Hammond, III, Bounding the Number of Internal Rates of Return, Technical Report,

Harvard Business School, Boston, Mass., 1976. 5. __ , Testing for Multiple Rates of Return by Simple Arithmetic, Technical Report, HBS 77-1, Harvard

Business School, Boston, Mass., 1977.

HARVARD UNIVERSITY BusmNEss SCHOOL, MORGAN 325, CAMBRIDGE, MA 02163.

SUMMING AN ALTERNATING SERIES

RICHARD JOHNSONBAUGH

Introduction. Isaac Asimov in a recent book [1, pp. 78-80] contrasts the sixteenth-century approximation for v, 355/113, with Leibniz' series for v,

00

sr= 4( - 1)'+ 1/(2k - 1), (*)

given in 1673. Asimov asks how many terms of (*) one must take in order to improve the estimate 355/113.

More generally, one can ask about the accuracy one obtains in approximating the sum of a convergent alternating series 21(- l)k+ 'ak by the partial sum ,7( -_l)k+ lak. We shall derive elementary, but useful, estimates in Section 1. In Section 3 we will use more sophisticated methods to derive other estimates.

Obviously, Leibniz' series (*) converges very slowly. Thus it is impractical to compute the sum of this series by simply adding up the terms. In Section 2 we will develop methods of approximating accurately the sum of a convergent alternating series E:(- l)k+ lak.

In [4] Boas investigated the questions raised above primarily for series of positive terms. Calabrese [6] and Pinsky [9] considered some of the above questions for the alternating series.

1. Elementary remainder estimates. We first fix some notation. Throughout this paper, if 7: 'bk is a convergent series, we shall let Rn =2 E'+ Ibk the error or remainder in approximating the actual sum by E2nbk.

If {bk) is a sequence, we will let Abk= bk-bk+1, the first difference of the sequence {bk).

Successive differences are defined inductively: Ai+lbk= A(Abk). We also set Aobk= bk.

Throughout this section we shall assume that {ak) is a positive, decreasing sequence with limit zero. The standard calculus results (see, for example, [10, p. 587]) tell us that in this case 2(_ I)k+ lak converges, {IRnl} decreases, and that if the sum of the series is denoted by L, then

? < Rnj I

-)nRn = (_ l)n+ 1 2 ( _ )k+ lak- L]

We take as our starting point the easily verified identity n -n-I

I _ )k+lak= al/2 +(1/2) 2 (_-I)k+IAak I( )n+l a /2 (1.1)

Richard Johnsonbaugh received his Ph.D. from the University of Oregon in 1969 under Bertram Yood. From 1969-72 he served as Chairperson of the Mathematics Department at Morehouse College, and from 1972-77 he served as Chairperson of the Mathematics Department at Chicago State University. He has held visiting appointments at the University of Victoria and the Argonne National Laboratory. His research interests lie in the areas of analysis and topology. He is the coauthor with W. E. Pfaffenberger of the forthcoming text Foundations of Mathematical Analysis. -Editors

638 RICHARD JOHNSONBAUGH [October

This identity is of a type studied extensively by Berndt and Schoenfeld [3]. Taking the limit as n-+oo, we see that the series in brackets [] converges to L also. Now subtracting L and multiplying by (- 1)"+' we obtain

' n JR1 | ) =_ n+1 1z(_ 1)k ak_ L

li 1 (1.2)

{((I)n [a/2+(1/2) I(1) ak aLI} + an/2.

If {Aak) decreases, the term in braces {) is negative and so JRnI <aqn2. This is Calabrese's result.

THEOREM 1.1 (Calabrese [6]). If {Aak} decreases, then IRnI an/2.

COROLLARY (Pinsky [9]). If {Aak) and {A2ak) decrease, then I Rntl < Aan /4,

where Rn' is the remainder for the series

al/2 +(1/2) 2 (-) k+I Aak. (1.3)

Proof. Apply Theorem 1.1 to the series (1.3).u

Pinsky [9] noted that if sn denotes the nth partial sum of the series (-l1)Y`ak, then the nth partial sum of the series (1.3) is (sn + Sn_- )/2. Thus the Corollary above gives an estimate for the accuracy obtained by approximating the sum of the original series by the averages of its partial sums.

Shohat [12] in 1933 obtained the series (1.3) for the special case ak = 1/(2k - 1) (which is 1/4 of the Leibniz series (*)) by a much different and more complicated method than that given above. Unfortunately, his method is not sufficiently general. (For example, it gives no useful information for the series j(- I)k+ I/k.)

An identity similar to (1.1) is

> (-1)k+ a[a/2+(1/2) (_) k+ ak] +(1)n+ an+1/2. (1.4)

Again taking the limit as n-*oo, the series in brackets [ ]converges to L. Again subtracting L and multiplying by (- l)n+ 1, we obtain

(1.5)

I -)n+l a,l/2 +(1/2)2(-)k+l/\ak -L + +an+, /2.

This time if {Aak } decreases, the term in braces {} is positive so an+ 1/2<1 Rn . This result has been noted before ([5]). The corresponding corollary is that if {A2ak) decreases, then Aan+1/4< I Rnl. Combining these above results we have theorems giving both upper and lower estimates for the original series and series (1.3).

THEOREM 1.2. If {ak) and {Aak) decrease, then

an+ 1/2 < IRnI <an/2.

COROLLARY. If {Aak) and {M2ak) decrease, then

Aan+ 1/4 < IRnl < Aan/4.


We may extend these arguments to obtain increasingly sharp estimates. We state our results as a theorem.

THEOREM 1.3. If j(- l)k+ lak converges and ({Aak} decreases, then i i z /i- 'a"+ 1 /2i < iRni <anl2

- I /Ai- anl2i.

i=l i=2

Proof. In identity (1.4) replace n by n-I and ak by Aak, then substitute into equation (1.2) to obtain

n-I IRnI lI)

I aJ)2+iaj/4+(I14) 2 I)k+ 1 5 2a -L]

+ an/2 - Aan/4.

Repeating this procedure j-2 times, we obtain

Rnf= (_1)n+l 2 Ai\al/2 +(1/2j) 2i (- k) Aak-LI?

+a/2_ 2 /vi-l a /2i. i=2

Since the term in braces () is always negative, the right inequality follows. The left inequality is derived similarly beginning with identity (1.5). U

Now letf be a real-valued, differentiable function defined on [1, oo). There is a close relation between the difference Af and the derivative f'. Indeed, by the mean-value theorem, we have

Af(k) = - [f(k + 1) -f(k)] =-f'(() for some ( E(k, k + 1). It follows that if f' is increasing, then {Af(k)) is decreasing. Similarly, if f" exists everywhere, A2f(k) =f"(() for some ( E(k, k + 1). Thus (A2f(k)) is decreasing if f" is decreasing. Therefore, in order to assure that {Ajf(k)) decreases forj = 0, 1,2,..., it suffices thatf is decreasing, f' is increasing, f" is decreasing,.... This latter condition is fulfilled if f' is negative, f" is positive, f"' is negative,.... These conditions are satisfied for many of the functions f defining alternating series, especially those involving terms like l/xP and ln x. A sequence { ak) for which ({Aak) is decreasing forj =0, 1,2,... is called completely monotone (see [13]). We see that if f(j)(t)f(U+')(t)<0 for j=0, 1,2,... and for tE[l, oo), then {f(k)) is completely monotone.

As an illustration of the use of these remainder estimates, let ak = 1/(2k - 1). Suppose we wish to find the smallest n for which IRnI < I X 10-4. (This precision is often referred to as four-place decimal accuracy.)

By the criteria above, {ak) is completely monotone, so that Theorem 1.3 applies. Using the estimate IRnI <an/2, we solve an/2 < I X 10-4, to obtain

5000.5 = (1I O+ 1)/2 < n. This inequality suggests taking n = 5001. Using the estimate an+ /2 < IRnI we find that

4.9995 x 10-5 = a5001/2 < IRoooI and

I X l0-4 < 5.0005 x 10-5 = a5000/2 < IR49I.

Recalling that { R;) decreases, we see that the desired n is either 5000 or 5001. To decide between these two values we must employ a sharper estimate.


We use Theorem 1.3, with j= 3, to obtain

1R50001 <a50001/2 -Aa50o/4-2a5OOO/8 =4.99999995 X i0-5 < I X 1-04.

Therefore n = 5000 is the desired value.

TABLE 1

(1) J,*(-l)k+1/ln(ln(k+2)) (2) jm(- 1)k+I/In(k +1) (3) j1 (_ I)k+l/kP

(4) 2 (-1)k?l1nk/k (5) c(-1)k+l/(2k-1) (6) o w(- )k (7) oo(- I)k/k!

(8) 100(_ l)k+l/kkk Series Sum Number of terms required to calculate the sum

with absolute error less than (1 /2) x 10-2 10-1lo 1-100-loo 10

1 8.74955124 T(l.17x 1043) T3(9.7) T3(99) T3(1000)

2 .924299897 2.688 X 1043 T(4.4x 109) T2(00) T2(1000)

3(p=.01) .5022548581 lwo' 100 10'0 000 l0100.0

3(p =.5) .6048986434 ioP 1020 l0O 102000

4 -.1598689037 648 2.7x 1011 2.4x 10102 2.4X 101003

3(p= 1) .6931471805 100 1010 10100 1o0'

5 .7853981633 50 5x109 5x1099 5x1099

6 10/19 45 220 2186 21,855

7 .3678794411 6 15 71 451

8 .9375 2 2 3 4

Note: T(x)= T,(x)= 10X', T,(x)= T(T,, I(x)).

Table 1 gives the number of terms required to approximate various sums with particular degrees of precision. The series are arranged roughly in order of their rates of convergence.

The next section is devoted to a discussion of how to calculate the sum of an alternating series. Of course some of the sums in Table I are known in closed form. For example, the sum of series (3) (p= 1) is ln 2. The sums of the rapidly converging series can be calculated by direct addition. The interesting sums to approximate are those given by series such as (1) or (2) where the sum is not known in closed form and where the series converges so slowly that it is not feasible to directly add the terms.

We close this section by answering Asimov's question quoted in the introduction, which is to find the least n for which

IRnl < 355/113- 7T,

where Rn is the remainder for the series 00( - )k+ lak with ak = 4/(2k - 1). Letting E =355/113-o , we approximate E by 2.667641890x 10-7. Solving a,/2 <E we obtain

n>3,748,629.5. Now for n=3,748,630, we have jR"j <a,/2<e. For n=3,748,629, we have e < 2.667641956 x 10-7 = a"+ 1/2 + Aa + 1/4 < I R. 1. Thus we must take a minimum of 3,748,630 terms of E(- 1)k+ ak to improve 355/113 as an approximation to 7.

Obviously summing the series 14(- l)k+l '/(2k-1) directly is an impractical means of approximating ir. Recent approximations of 7T to several thousand decimal places have been obtained by using certain arctangent relations, such as Machin's formula,

-r/4=4 arctan(l/5) - arctan(l/239),

and the Gregory series,


00

arctan x= E (-l)k+Ix2k-1/(2k- 1), 1

to evaluate the arctangents. (See [2] and [14] for details and a history of the approximation of so.) Apparently, the record approximation to so is held by J. Guilloud, who computed one million decimal digits. A new algorithm for computing '7 published in 1976 by Salamin [11] promises to surpass previous methods. Salamin's formula is based on Gauss' method for calculating elliptic integrals and an elliptic integral relation of Legendre. A proof of Salamin's main result can be given using only results from elementary calculus. Guibas and Simonyi are working on a program to compute so to 67 million binary digits using Salamin's algorithm. This is equivalent to approximately 20 million decimal digits.

2. Summing the alternating series. The usual method of summing a slowly converging series >ak is to replace it with a series 2bk which has the property that its partial sums sn = i7bk, for small values of n, give good approximations to the sum of the series 2ak. The series L:bk may be either convergent or divergent. Indeed, perhaps the most powerful method for summing a slQ)wly convergent series involves a divergent series lbk. However, at first we shall follow what is perhaps the most obvious route, simply replacing a,ak by a convergent series 2bk which has a more rapid rate of convergence.

Let {ak) be a positive, decreasing sequence with limit zero. As usual let Rn denote the remainder for the series

00

2 (-l)k+ lak. (2.1)

In Section 1 we showed that the series 00

al/2 +(1/2) (1) k+

^k (2.2)

has the same sum as 1 (-l)k +'a. Let Rn' denote the remainder for this series. Theorem 1. I and its Corollary show that if {Aak) and {MAak) decrease, the remainders satisfy

IRn l<an/2, IRn'I <Aan/4. Therefore, replacing series (2.1) with series (2.2) improves the rate of convergence by at least a factor of two. Table 2 gives a comparison of the actual rates of convergence of the series (2.1) and (2.2) for ak= 1/k.

Of course this process may be repeated. That is, we may replace the alternating series E 0((-l)k+ Aak in (2.2) by Aaj/2+(l/2)20(- l)k+ 'ak. Substituting back into (2.2), we obtain

aj/2+Aa1/4+(l1/4) c (-l)+ Ma,. (2.3) 1

Letting R,," denote the remainder for this series, we find that if {A2ak) and {A3ak) decrease, then

JR,"| <" an/8.

We again achieve at least a twofold rate of increase in convergence in replacing series (2.2) with (2.3). This process obviously may be continued to obtain remainders R"'n,.... Table 2 also compares Rn and Rn" for l(- l)k+l/k.

If we do continue this process, we obtain 00 n-I 00 I

ak= a/2k+l+(l/2n) 2 (-l)k+lAn (2.4) 1 0 k-i

If :( -l)k+ lak converges, the series (1/2n)E: ( - l)k 1Atak converges and equation (2.4) holds. This result follows from equation (1.1). In fact one need not assume that {ak) is monotone or even that ak >0, although most of our series will possess these properties.


8~~~~~~~~~~~~~~~~o o N 00 e,,S i o

,1 C S 34 8 i? 8 :g 80 ;i x i 9 x \i x P% 4 i) ~0

+ - ? x > 8 8o ̂8 i ,8> 3 8 g x ^ x~~~~~~~~~~~~- t cs Z ?, t l l ~~~~~~~~~~~~~~~~~~~~~~~-l | | o |o;R

4 H C ̂ ^ ? ̂ ? ̂ ? N ? t o tO o~~~~~~~~~~"4 14 - R + ?~~~~~~~~~~~~~~c x eniow_W8 x ef x

_? + 3 H ?n > XD O sD 8.; D O 8 s O en 0

S~~~~~~~~~~~c 00 tn _0 cn v L ?|WE _ cO x d t t 8 ^ : s XD t o t~~~~~~~~~~n M x CC) x

S.~ ~ ~ ON tWuu?'


We should be tempted to let n->oo in equation (2.4) and use

I Aka/12"k1 (2.5) 0

as an approximation to the original series (2.1). This will be permissible if 00

n--*oo k=1

This is, in fact, true under the sole assumption that series (2.1) is convergent. This result is due to Euler and series (2.5) is called the Euler transformation of series (2.1).

THEOREM 2.1 (Euler [8, pp. 244-246]). If 2( I-j)k+ 'ak converges, then 1OAkaI/2k+l converges also and has the same sum.

In Table 2 we compare series (2.5) with series (2.2) and (2.3) and we note that series (2.5) converges most rapidly.

To obtain an estimate for the remainder of the Euler transformation (2.5), we return to equation (2.4). Letting RnE denote the nth remainder of the Euler transformation, we see that

00

RIn (I1/2n) I (- i) k

Iak. k=1

Therefore, if {Anak) decreases, RnE < An al/2n.

For example, using this estimate, we can show that only 22 terms of the Euler transformation of Leibniz' series for so are needed to improve 355/113 as an estimate for "r.

One can approximate the alternating series j(_- )k+ lak very accurately by directly summing N terms, then using the Euler transformation to approximate the remainder 2 '+ ( -l)k + lak. In Table 2, we approximate j(-_ )k+ 'lk by

100 n

I,(1)k I/ k + 2 Aka,Ol /2 k+ 1 1 0

for various n. Great accuracy is obtained. The Euler transformation is most useful when the differences Aka, are easily computed. In

general the most successful method to sum a slowly converging alternating series is to obtain an integral representation for the series. This we can easily do by beginning with equation (1.1). I am indebted to Professor Ralph P. Boas, Jr., for pointing out the intimate connection between equation (1.1) and our integral representation.

Assuming that f and its derivatives are continuous on [1, oo), we may write n n-I

(-I)k+ f(k) [f(l) + (I) f(n) /2+ (1/2) I (-1) Af(k)

=[f(l) + (-)+ 'f(n) ]/2 + f f'(t)X,(t)dt, (2.6)

where Xl(t)= - 2 on [1,2),[3,4),[5,6),... and Xl(t)= + I on [2,3),[4,5),[6,7) .... (See Fig. 1.) Equation (2.6) is a special case of Boole's formula.

To improve equation (2.6) we use integration by parts. It is convenient to choose the indefinite integral X2(t) of Xl(t) to be periodic of period 2. (See Fig. 2.) We obtain

ff'(t)XI(t)dt=f'(t)X2(t) -1f"(0X2 d

=f'(n)X2(n) -f'(1)X2(1) - f "(t)X2(t)dt


y

20~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~om" _.k

2

y=Xl(t)

Fio. 1

y

4 ? )*

y =X2(0)

FIG. 2

y

'l tO

1F6

Y =X3(t)

Fio. 3


=f'(n)(- _) ln+1 C2-f'(l) C2- f"(t)X2(t)dt,

where C2 = (-l)"+ 'X2(n) = X2(1) = 1/4. Substituting back into equation (2.6) we have n

I)+'(k) [f(l) +(_ I)n f(n) ]/2 + C2[(- I)n+ lf (n) -f (1 n

- f"(t)X2(t)dt.

If we again integrate by parts and again choose the periodic, indefinite integral X3(t) of X2(t) (see Fig. 3), we obtain

I(k) 'f(k) [f( l) + (_-I)n +f(n)] /2 + C2 [ ( I)n + f'(n) -f'(l)] n

+ ff"(t)X3(t)dt

since this time X3(1) = X3(n) = 0. Repeated integration by parts gives

n m

2(l1)k+ f(k) [f(l)+( 1) +f(n)]/2+ 2 C2k[(-1) f((k l)(n) 1 1

]+ ff(2m+)(t)X2m+t(t)dt, (2.7)

where the constant Ck+1 is chosen so that Xk+ 1(t) is the periodic indefinite integral of Xk(t). We shall always have Xk(l)=Xk(n)=O for k odd and k >3. (An empty sum, YmI for m <1, is interpreted as zero.)

If we take f(t) = t2m, the integral in equation (2.7) is zero. If we then take n = 2, we can develop a formula from which the C2k can be determined recursively. It is also possible (see [4]) to write the numbers C2k in terms of the Bernoulli numbers B2k. The relationship is given by the equation

22k- 1 C2k= (2k)! 2k-

Some Bernoulli numbers are tabulated in [7]. The first few Bernoulli numbers are

Bl=- B I B 1 B 1 8 2' B2= 6 4 -30' B6= 42 30

B 5 691 66 B12 =2730

The functions Xk(t) can be related to the periodic extensions of the Bernoulli polynomials. Details are given in [4].

It is tempting to let m->oo in equation (2.7) to obtain an infinite series on the right to approximate the original series on the left. Unfortunately, the resulting series is usually divergent. Therefore, we proceed differently.

First, assume that j(- l)k+ 'f(k) converges and that the derivatives f(k)(t) converge to zero as t-*oo. We may then let n-*oo in equation (2.7) to obtain

a: m 00

Replag (

1))k+ =fl/2-2

c2J(2k

- 1)()+ f(2m+l)()X

(t)dt 11 1

Replacing the lower index 1 by N we have

N 1 N


We shall show in Section 3 how to obtain accurate estimates of the integral IN=

f NXf(2m+ 1)(t)X2m+ I(t)dt. Thus, we can use equation (2.8) to approximate the sum ,(- l)k lf(k). This approach is similar to the situation we encountered in using the Euler

transformation. We can sum a series to N terms, then use formula (2.8) to approximate the remainder.

For example, if f(k)= 1 /k, m = 1, and N= 101, we obtain 00 101

I -)k+ l/k I

-)k+ l/k+ 1/(2 -101) + 1/(4 -1012) + Iol. 1 1

Neglecting Ilol, this sum is .6931471817, which differs from the true value by approximately 1.2 x I0-9. For larger values of m, we obtain more accuracy. (See Table 2.)

3. Additional remainder estimates. In this section we will first derive estimates for the remainder IN= f5 f(2m+ )(t)X2m+ 1(t)dt appearing in equation (2.8). We will then be able to justify the sums given in Table 1 and explain how they were derived. We will conclude by obtaining estimates for the remainder E +((- l)k+ f(k) of the alternating series in terms of the derivatives of f. These formulas will complement the formulas given in Section 1 where the remainder is given in terms of the differences Akf.

Throughout this section we shall assume that f is positive and that the derivatives f(k)(t) tend to zero strictly monotonically as t->oo. It follows that f(2k - ')(t) is negative and increases to zero as t--oo.

We note that Xk(t+ 1) =-Xk(t) for k= 1,2,3 (see Figures 1, 2, and 3). Using induction we can show that X2k+ l(t + 1)= -X2k+ l(t) for k = , 1, 2..... We also note that Xl(t) < 0 for t E[1,2) and X3(t) > 0 for t E[l, 2). Using induction we can show that X2k+ I(t) < 0 for k even and t E[1, 2) and X2k+ I(t)> 0 for k odd and t E[l,2). Clearly, no function Xk(t) is identically zero.

First, suppose that N is odd. If m is odd, in view of the above discussion f N+2ft+I) (t) X2m + I (t) dt

N

= f N+ 'f(2m + ')(t)X2m + I(t)dt + JN 2pm+ 1)(t)X2m+ 1(t)dt N N+1

= fN+ 'f(2m+ 1)(t)X2 m (t)dt _ fN+ (2m+ 1)(t + I)X2m+ ,(t)dt N Nt

jN+1 [ f(2m+l)(t) _f2m+l)(t+ l)]X2m+l(t)dt <0 N

since f(2m+ l)(t) - f(2m+ 1)(t + 1) < 0, X2m+ I(t) > 0 on [N,N+ 1), and X2m+ I(t) is not identically zero. Thus,

Jf 2f2m+ Q)(t)X2m+ (t)dt <0

for odd i. Therefore f N+2Y(2m+')(t)X2m+ (t)dt<0, for i= 1,2,.... Taking the limit as i->oo we obtain

IN = f f(2m+

1)(t)X2m+ ,(t)dt <0. N

If N is odd and m is even, the argument is the same except that X2m+ I(t) < 0 on [N,N+ 1). In any case we have (- l)mIN >0. If N is even we get a change of sign so that, in general,

(_I)m +N+ 11N>o.

Fix N. Letting Jm = f 5fP(2m+ ')(t)X2m+ I(t)dt, Jm and Jm+ I have opposite signs. Therefore Jm and Jm-Jm+ l have the same sign. Thus IJmI < IJm-Jm+ I1. However

Jm-Jm+, 1=-(- 1) NC2mN+2 f (2m+ ')(N).


It follows that we may write A f 00m+ l)()X2m+ l(t)d = Jm =-71(1) +C2m+2f (2m+ ')(N), f~(2m + ')(t) X2 I (t) dtJ=j

I N N

for some q, 0<7 < 1. We summarize our results as Theorem 3.1.

THEOREM 3.1. Suppose f is positive and that f and all of its derivatives f (k)(t) tend to zero strictly monotonically as t->oo. Then

oo m 2 (_ )k+lf(k)( 1)N+1 f(N)/2 2 C2J(2k

- 1)(N)-C2m +2 f (2m+ 1)(N)

N I

for some q, 0 <q < 1. It can be shown that the numbers C2k alternate in sign. Therefore, the series above on the

right is an alternating series. Even though {I C2kf(2k- ')(N)I} is rarely a decreasing sequence, the series behaves like an alternating series '((- l)k+ lak, where {akl decreases, in that the remainder in stopping at any term is less than the absolute value of and has the same sign as the next term. Tlhis is the conclusion reached in introductory calculus texts for the alternating series

(- l)k+ lak, where {ak} decreases, (see, for example, [10, p. 587]). In the previous section we estimated

00

i (- _) k+1 f(k)>f(101)/2 -f'(101)/4=.6931471817, 101

where f(k)= I/k. We may now employ Theorem 3.1 to find that the error is at most the absolute value of and has the same sign as

- C4f.."'(101)=f"'(101)/48= - 1.201225430X 10-. This value is in very close agreement with the actual error (see Table 2).

We used Theorem 3.1 to sum the slowly converging series in Table 1 whose sums are not known in closed form. For example, to sum 2O Il)k+'/k01, we first summed 500 terms to obtain 15O(- l)k+ l/k-l = .03238675800. We then estimated

00

2 ( I)k+ '/k 01

f(501)/2 -f'(501)/4= .4698681001. 501

The error in this last approximation is less than the absolute value of and has the same sign as f"'(501)/48= -3.16x 10-12. Therefore, the sum joo(- l)k+l/k.01 is .03238675800+.4698681001 =.502254858, correct to nine places. As Table 1 shows,, it is impractical to sum this series by direct addition. The other sums in Table 1 were computed similarly.

To derive error estimates for the remainder Rn = E'+ I(_- )k+ 'f(k), we write

Rn =Rn_ I + (- I)nf(n)

=(-I)f(n)/2+ (- )n [- C2kfJ(2k -)(n) -qC2m+2f(2m+ 1)(n)] (3.1)

for some 7, 0 <71 < 1. Taking m = 0, we have

Rn I -)nf(n)12 + (- I)n7f'(n)14.

Thus

IRnl = (- )nRn = f(n)/2 + qf '(n)/4 <.f(n)/2

since f'(n) <0. We again have Calabrese's result (Theorem 1.1). We can obtain further error estimates by taking more terms in equation (3.1).

THEOREM 3.2. Suppose f is positive and that f and its derivatives f(k)(t) tend to zero strictly monotonically as t->oo. Let Rn = 2`+(_- l)k+ lf(k) denote the nth remainder of the alternating

648 EMIL GROSSWALD [October

series. Then

IRnI =f(n)/2 + , C2Jf (2'- l)(n) + qC2m+2f (2m+ )(n) 1

for some q, 0 <,q < 1. Furthermore, 2i+1 2i

f(n)/2 + C2kf (2k- l)(n) < I Rnl I f(n)12 + > C2kf (2k- 1)(n) 1 1

for i=O, 1,2,....

Proof. The equation is merely another way to write equation (3.1). Since C2m >O if m is odd and C2m < 0 if m is even, the inequality follows immediately. m

References

1. I. Asimov, Asimov on Numbers, Doubleday, Garden City, N.Y., 1977. 2. P. Beckmann, A History of Pi, Golem Press, Boulder, Colo., 1970. 3. B. C. Berndt and L. Schoenfeld, Periodic analogues of the Euler-Maclaurin and Poisson summation

formulas with applications to number theory, Acta Arith., 28 (1975) 23-68. 4. R. P. Boas, Jr., Partial sums of infinite series, and how they grow, this MoNrTLy, 84 (1977) 237-258. 5. _ , Estimating remainders, Math. Mag., 51 (1978) 83-89. 6. P. Calabrese, A note on alternating series, this MONTHLY, 69 (1962) 215-217. 7. L. B. W. Jolley, Summation of Series (2nd rev. ed.), Dover, New York, 1961. 8. K. Knopp, Theory and Application of Infinite Series, Blackie, London and Glasgow, 1928. 9. M. A. Pinsky, Averaging an alternating series, Math. Mag., 51 (1978) 235-237. 10. D. F. Riddle, Calculus and Analytic Geometry (2nd ed.), Wadsworth, Belmont, Cal., 1974. 11. E. Salamin, Computation of X using arithmetic geometric mean, Math Comp., 30 (1976) 565-570. 12. J. A. Shohat, On a certain transformation of infinite series, this MoNTHLY, 40 (1933) 226-229. 13. D. V. Widder, The Laplace Transform, Princeton University Press, 1941. 14. J. W. Wrench, Jr., The evolution of extended decimal approximations to xT, Math Teacher, 53, (1960)

644-650. DEPARTMNT OF MATHEMATICS, CHICAGO STATE UNIVERSITY, CHICAGO, IL 60628.

RECENT APPLICATIONS OF SOME OLD WORK OF LAGUERRE

EMIL GROSSWALD

1. Introduction. Edmond Laguerre (1834-1886) is rightly considered as one of the foremost mathematicians of his time. He was a forerunner of Hadamard in the study of entire functions; the "Laguerre polynomials" are an important tool in several branches of pure and of applied mathematics, and Laguerre is also often quoted for his contributions to geometry ("theory of cycles"), algebraic equations, and continued fractions.

Nevertheless, he rates only four half-lines in the 1972 edition of the Petit Larousse [15], and his name is not even mentioned in such excellent surveys as [8], [18], [19], and [14]. To my surprise, not only does Laguerre not rate an entry, but he is not even mentioned under some other heading, in the Encyclopaedia Britannica (at least not in its 1954 edition) [5]. There is one brief mention of Laguerre in the four-volume World of Mathematics [12] in connection with the

Emil Grosswald, a native of Romania, has a master's degree in mathematics and electrical engineering from Bucharest, a degree in electrical engineering from Paris, and a Ph.D. in mathematics (directed by H. Rademacher) from the University of Pennsylvania (1950). From 1939 to 1948 he lived successively in France, Cuba, and Puerto Rico. Except for a few years at the Institute for Advanced Study, the University of Paris, and the Technion, he has spent most of his professional life at the University of Pennsylvania and (since 1968) at Temple University. His main interests are in number theory and algebra.-Editors.

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