Convergence Acceleration of Alternating Series

Transcription

1 Convergence Acceleration of Alternating Series Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier CONTENTS The First Acceleration Algorithm Two Flavors of the Second Algorithm Applicability Extension to Certain Nonalternating Series References We discuss some linear acceleration methods for alternating series which are in theory and in practice much better than that of Euler Van Wijngaarden. One of the algorithms, for instance, allows one to calculate P ( 1) k a k with an error of about n from the first n terms for a wide class of sequences fa k g. Such methods are useful for high precision calculations frequently appearing in number theory. The goal of this paper is to describe some linear methods to accelerate the convergence of many alternating sums. The main strength of these methods is that they are very simple to implement and permit rapid evaluation of the sums to the very high precision (say several hundred digits) frequently occurring in number theory. THE FIRST ACCELERATION ALGORITHM The typical series P we will be considering are alternating series S = 1 k= ( 1)k a k, where a k is a reasonably well-behaved function of k which goes slowly to as k! 1. Assume we want to compute a good approximation to S using the rst n values a k. Then our rst algorithm is: Algorithm 1. Initialize: d = (3 + p 8) n ; d = (d + 1=d)=; b = 1; c = d; s = ; For k = up to k = n 1, repeat: c = b c; s = s + c a k ; b = (k + n)(k n)b=((k + 1 )(k + 1)); Output: s=d. Keywords: Convergence acceleration, alternating sum, Chebyshev polynomial. AMS Classication: 11Y4 This algorithm computes an approximation to S as a weighted sum of a ; : : : ; a n 1 with universal rational coecients c n;k =d n (= c=d in the notation of the algorithm; note that both c and d are integers). For instance, for n = 1,, 3, 4 the approximations given c A K Peters, Ltd / $.5 per page Experimental Mathematics 9:1, page 3

2 4 Experimental Mathematics, Vol. 9 (), No. 1 by the algorithm are a =3; (16a 8a 1 )=17; (98a 8a 1 +3a )=99; (576a 544a a 18a 3 )=577; respectively. The denominator d n grows like 5:88 n and the absolute values of the coecients c n;k decrease smoothly from d n 1 to. Proposition 1 below proves that for a large class of sequences fa k g the algorithm gives an approximation with a relative accuracy of about 5:88 n, so that to get D decimal digits it suces to take n equal to approximately 1.31D. Notice that the number of terms and the time needed to get a given accuracy are essentially independent of the particular series being summed, if we assume that the a k 's themselves are either easy to compute or have been precomputed; on a Sparcstation 5 using Pari, for instance, the computation of S to 1 or 1 decimal digits requires about.1 or 6 seconds, respectively. The algorithm uses O(1) storage and has a running time of O(1) per value of a k used. Proposition 1. For integers n k set d n = (3 + p 8) n + (3 p 8) n (1) and kx! c n;k = ( 1) k n n + m d n m n + m m m= nx = ( 1) k n n + m m : () n + m m m=k+1 Assume that the a k are the moments of a positive measure on [; 1] and set Then S = k= ( 1) k a k ; S n = js S n j S d n n 1 X k= S (3 + p 8) n : c n;k d n a k : R 1 Proof. By assumption, a k = xk d, where d is a positive measure. Then S = k= ( 1) k a k = x d; the interchange of summations being justied by the positivity. Let fp n (x)g be a sequence of polynomials such that P n has degree n and d ~ n := P n ( 1) 6=. Set n 1 P n ( 1) P n (x) X = ~c n;k x k : 1 + x k= Dene Sn ~ as S n in the proposition but with d ~ n and ~c n;k instead of d n and c n;k. Then ~S n = say, with 1 P n ( 1) R n = P n ( 1) P n (x) 1 + x d = S R n ; P n (x) d ; (3) P n ( 1)(1 + x) and by virtue of the positivity of d we can estimate the \error term" R n by jr n j M n jp n ( 1)j x d = M n jp n ( 1)j S ; where M n is the supremum of jp n (x)j on [; 1]. It follows that M n =jp n ( 1)j is an upper bound for the relative error made by approximating S by ~ Sn. We now choose for P n (X) the polynomials dened by P n (sin t) = cos nt; (4) so that P n (x) = T n (1 x) where T n (x) is the ordinary Chebyshev polynomial. Clearly M n = 1 and ~d n = P n ( 1) = d n with d n as in (1). The recursion P n+1 (X) = (1 X)P n (X) P n 1 (X) implies by induction the explicit formula nx P n (X) = ( 1) m n n + m m X m ; (5) n + m m m= so ~c n;k = c n;k with c n;k as in (). This completes the proof of the proposition. Remarks. 1. The method implicit in Proposition 1 is well known in numerical analysis under the heading of \Pade type approximation" [Brezinski 198; Eiermann 1984; Gustafson 1978/79; Wimp 1981]. As we mentioned above, we are concerned with calculations to a high degree of accuracy, where the number of digits gained per number of steps, and the amount of storage required, are crucial. Thus our

3 Cohen, Rodriguez Villegas, and Zagier: Convergence Acceleration of Alternating Series 5 emphasis is dierent from that in numerical analysis, where one usually works in xed, and relatively low, precision. The implementation of Algorithm 1 is good in both respects.. The classical algorithms found in the literature (see [Press et al. 1988]) are Euler's method or Euler{ Van Wijngaarden's method. These can be shown to correspond respectively to the polynomials P n (X) = (1 X) n with convergence like n and polynomials P n (X) = X a (1 X) b with a + b = n dependent on the particular sequence, with convergence like 3 n for a = n=3. Note that a direct implementation of the algorithm given in [Press et al. 1988] needs a lot of auxiliary storage if we want high accuracy, while our method does not. 3. Algorithm 1 computes \on the y" the coecients of the polynomial (P n ( 1) P n (X))=(1+X), where P n (X) = T n (1 X). Equivalently, we could also compute on the y only the coecients of the polynomial P n (X) itself and use the partial sums of the alternating series instead of the individual terms, using nx k= c n;k a k = nx m=1 n n+m n+m m X m m 1 k= ( 1) k a k : This can be particularly useful when the sequence of partial sums is naturally given, and not the a k themselves, as in the continued fraction example mentioned at the end of this paper. 4. The hypothesis that the a n 's are moments of a positive measure on the interval [,1] is a well known one, and is equivalent by a famous theorem of Hausdor [193] to the total monotonicity of the sequence fa n g, in the sense that for each xed k, the sequence f k a n g of the k-th forward dierences of fa n g has constant sign ( 1) k. 5. In addition to this, trivial modications are possible. For example, one can replace the step d = (d + 1=d)= by the simpler one d = d=, since this modies the nal result by a negligible amount, but leaves d as a nonintegral value. We could also immediately divide by d (initializing b to 1=d and c to 1). In our implementation each of these modi- cations led to slower programs. 6. We proved convergence of the algorithm (at a rather fast geometric rate) under the above condition. However, examples show that it can be applied to a much wider class of series, and also, as is usual in acceleration methods, to many divergent series. 7. The choice of the Chebyshev polynomial can be shown to be close to optimal if we estimate the remainder term R n crudely as we did above. On the other hand, as we will see below, for a dierent class of alternating series, we can estimate R n more precisely and nd much better polynomials P n. The corresponding algorithms and their analysis seem to be new. 8. If the sequence a k already converges at a geometric rate, better algorithms are possible which are trivial modications of the ones presented in this paper. For example, if one knows in advance that ln(a k ) k ln(z) for some z 1, then in Algorithm 1 simply replace 3+ p 8 by z+1+ p z(z + 1) and multiply by z the right hand side of the recursion formula for b. The convergence is in (z p z(z + 1)) n, and thus faster than the direct application of Algorithm 1. Similar modications are valid for the other algorithms of this paper. The details are left to the reader. To illustrate Algorithm 1, we give a few examples. Each can also be treated individually using specic methods. Example 1. By taking logarithms we can compute the product A := 1Y n=1 (1 + 1 n 1 ) (1 + 1 n ) = 1: : : : rapidly to high accuracy. The product is slowly convergent and the gamma function is hard to compute, so our method is very useful here. In fact, as we will see below, Algorithm B is even better in this case. Example. Similar remarks apply to the sum B := n= ( 1) n Li n = 1: : : : where Li (x) = P 1 k=1 xk =k is the dilogarithm function. This sum arose in connection with the computation of a certain denite integral related to Khinchin's constant [Borwein and Crandall ].

4 6 Experimental Mathematics, Vol. 9 (), No. 1 Example 3. The Riemann zeta function can be calculated for reasonable values of the argument by 1 1 s (s) = n= ( 1) n (n + 1) s ; and we nd values like ( 1 ) = 1: : : : or even ( 1 + i) = : : : : : : : : i to high accuracy and very quickly (comparable to Euler{Maclaurin). Note that the latter example works even though P the coecients a k of our \alternating" series ( 1) k a k are not alternating or even real and do not tend to zero. The derivatives of (s) can also be calculated similarly, e.g. we can compute C 1 := n=1 ( 1) n log n p n = : : : : by Algorithm 1, and then use the identity C 1 = (1 p ) ( 1) + p ln ( 1 ) to deduce the value of ( 1 ) = 3: : : :. In a similar manner, we can compute C = n=1 ( 1) n log n n = : : : : by Algorithm 1, and then use the identity C = log ( 1= log ) to deduce the value of Euler's constant = : : : :. Moreover, as suggested to us by R. Sczech, we may even use Algorithm 1 to sum the divergent series C 3 = n=1 ( 1) n log n = : : : : ; recovering the known value 1 log( ) of the derivative of (1 1 s )(s) at s =. Note that in the rst two examples a k was in fact the restriction at points of the form 1=k of a function analytic in the neighborhood of, and so other techniques could be used such as expanding explicitly a k in powers of 1=k. However, in the last examples, this cannot be applied. Computing the constants C i for i = 1; ; 3 using algorithm 1 with n = 655 took about seconds in a Gp-Pari implementation on a 3 MHz computer running Linux. The relative errors were: for C 1, for C and 1 5 for C 3, not far from the bound 1=d of Proposition 1. If we make dierent assumptions on the sequence fa k g, we can use better polynomials than the Chebyshev polynomials. For example, we can reason as follows. Assume that d = w(x) dx for a smooth function w(x). Then taking P n (X) and d n as in (4) and (1) respectively and setting x = sin t, we get from (3) d n R n = = cos(nt) w(sin t) sin t 1 + sin t cos(nt) h(t) dt for a smooth function h(t). If we integrate twice by parts, we get d n R n = 1 4n ( 1) n c 1 + c dt Z 1 = cos(nt) h (t) dt; (6) 4n for some constants c 1 = h (=) and c = h (). If we multiply both sides of this equation by n, then the rst term 1 4 (( 1)n c 1 + c ) has period. This suggests replacing the polynomial P n (X) by the polynomial P (1) n (X) = n P n (X) (n ) P n (X): It is then easily seen that for the new remainder R n (1) we have d n R n (1) = O(1=n 5 ) instead of O(1=n ). Hence for our purposes P n (1) is a better polynomial than P n. Notice also that P (1) n (sin t) = 1 d sin(t) sin((n 1)t) : dt Continuing in this way, we dene a double family of polynomials P n (m) as the m-th dierence (with step ) of the sequence n m+1 P n (where we set P n = P n for n > since cos(nt) is an even function of n), and nd that for m > P (m) n (sin t) = 1 d dt m+1 sin(t) m sin((n m)t) The corresponding remainder term R (m) n d n R (m) n = O(1=n m+ ) satises (and even O(1=n m+3 ) if m is odd) as n! 1 for xed m, so we get better and better sequences of polynomials.

5 Cohen, Rodriguez Villegas, and Zagier: Convergence Acceleration of Alternating Series 7 As polynomials in X, these polynomials can be computed either from the formula giving the m-th dierence, i.e., P (m) n (X) = X rm ( 1) r m r (n r) m+1 P n r (X) (where we naturally set P n (X) = P n (X)) or by the formulas (X) = m+r 1 (D) r s X n m ( 1) X k k+r (1 X) n m 1 k+r k+1 P (m) n n m 1 k= where m + 1 = r + s with s 1, D = d d ; and = X(1 X) dx dx + 1 X: This suggests taking either the diagonal sequence or the sequence A n = P (n 1) n =( n 1 n!) (7) B n = P (m) n = (n m)(m + 1)! m (8) with m = bn=c, where the (unimportant) normalizing constants have been chosen so that A n () = B n () = 1 (other ratios between m and n could be considered and we will briey comment on this later). The rst few values are A (X) = 1; A 1 (X) = 1 X; A (X) = 1 8X + 8X ; A 3 (X) = 1 X + 54X 36X 3 ; and B n (X) = A n (X) for n 4. Note that the formulas giving the polynomials A n are particularly simple: we have and A n (X) = A n (sin t) = 1 n n! d n dt n sin n t ; (9) r (1 X) d r (r)! dx + (X X ) d dx if n = r + s with s 1. X r (1 X) r (1 X) s TWO FLAVORS OF THE SECOND ALGORITHM Algorithm 1 can be immediately generalized to any sequence Q n of polynomials, except that the coef- cients of Q n cannot be computed on the y, thus giving rise to the following family of algorithms. Algorithm Q. P n Let Q n (X) = b k= k X k. Set d = Q n ( 1); c = d; s = ; For k = up to k = n 1, repeat: c = b k c; s = s + c a k ; Output: s=d. In particular, applying this to the families Q n = A n and Q n = B n (dened in (7) and (8) respectively), we obtain two algorithms A and B which are of the same sort as Algorithm 1 in that they output an approximation S n which is a universal rational linear combination of a ; : : : ; a n 1. The values for n = 1 and n = are a =3, (16a 8a 1 )=17 as before, while those for n = 3 and n = 4 are (11a 9a a )=111 and (88a 168a a 51a 3 )=191, respectively (since A n = B n for n 4, the coecients are the same for both algorithms up to that point). We now analyze the speed of convergence of these algorithms. For Algorithm A we will nd that it is like 7:89 n for a large class of sequences fa k g and like 17:93 n for a smaller class, both better than the 5:83 n we had for Algorithm 1. For the same two classes of sequences, Algorithm B will be like 9:56 n and 14:41 n. In other words, depending on the sequence Algorithm A or B may be the better choice. On the other hand, unlike Algorithm 1, we do not have a quick way to compute the individual coecients c n;k in time O(1) each but must compute (and store) the whole polynomial A n (X) or B n (X). As a result, these algorithms require storage O(n) and time O(n ) instead of O(1) and O(n) as before. Thus they are inferior to Algorithm 1 if the numbers a k are easy to compute (like a k = 1=(k + 1)), but superior to it if the main part of the running time is devoted to the computation of the a k (as in Examples 1 or above), or in the extreme case when we only know a xed number of values a ; a 1 ; : : : ; a n 1 and want to make the best guess of the value of P ( 1) k a k on the basis of this data.

6 8 Experimental Mathematics, Vol. 9 (), No. 1 To state the result we need the constants dened by A = cosh A = 7: : : : ; B = e B = 9:55699 : : : ; A = A r = 17: : : : ; B = B p r = 14:48763 : : : : Here A = :75468 : : : and B = :5779 : : : are the unique real roots of A tanh A = L and B e B sinh B = L, respectively, where L = log(1 + p p ), and r = 4t = sin 4t = :736 : : :, where t = :65916 : : : is the unique root in [; =] of t + L = 1 t tan t. In addition, let CA be the union of the images in the complex plane of the two maps t 7! sin t (sin(t)=)(cos(t) i sin(t)) ; for t =, and let CB be the union of the images in the complex plane of the two maps t 7! sin t (r sin(t)=) r cos(t) i p 1 r cos(t) ; for t t = t (see Figure 1). Proposition. Assume that the coecients a k are given as moments, a k = w(x) x k dx: For Q = A or Q = B, let Rn Q = S Sn Q be the dierence between the true value of S and the output Sn Q of Algorithm Q. 1. If w(x) extends analytically to the interior of the region bounded by the Curve CA, then for Q = A or Q = B, jr Q n j is bounded by ( Q + o(1)) n.. If w(x) extends analytically to the whole complex plane (or only to the interior of the region bounded by the Curve CB), then for Q = A and Q = B, jr Q n j is bounded by ( Q + o(1)) n. In fact the proof will show that to get convergence like n Q we can allow singularities x m 1= (m ) at the origin (i.e., it is sucient if xw(x ) rather than w(x) itself be analytic). If w is analytic in a smaller region than those described in Proposition, then we still get exponential convergence of Sn Q to S, but with a worse constant. Here are a few simple examples. We set = Q and = Q with Q = A or Q = B, depending on whether Algorithm A or B is used. Example 4. Set a k = 1=(k + 1), S = log. Here w(x) = 1, so the proposition applies directly to give speed of convergence n. Hence in this case Algorithm A is better than Algorithm B, Example 5. Set a k = 1=(k + 1), S = =4. Here w(x) = 1 x 1=, so xw(x ) is analytic and we again get convergence n. Once again Algorithm A is better. Example 6. Set a k = 1=(3k+1), S = (log += p 3)=3. Here w(x) = 1 3 x =3 with a singularity at, so the convergence is like n. Hence in this case Algorithm B is better than Algorithm A. Example 7. Set a k = 1=(k + 1), S = =1. Here w(x) = log(1=x), again with a singularity at, so we get n convergence. The same applies to a k = 1=(k+1) s, where w(x) is proportional to log s 1 (1=x) and the convergence is again like n. Again Algorithm B is better. Example 8. Set a k = 1=(k +1), S = 1 +=(e e ). Here w(x) = sin(log(1=x))=x, again with convergence like n. Again Algorithm B is better. Proof of Proposition. We rst consider the case of Algorithm A. The rst thing we need to know is the asymptotic behavior of A n ( 1). According to Lagrange's formula, if z = a+w'(z) with ' analytic around a then dz X 1 da = d n da n ('n (a)) wn n! n= (1) (dierentiate [Hurwitz and Courant 199, p. 138, eq. (8)] with respect to a taking f(z) = z, for example). Choosing '(z) = sin z; w = u= and a = t in combination with (9) yields the identity n= A n (sin t) u n 1 = 1 u cos z and in particular n= z u sin z=t (11) A n ( 1) u n 1 = 1 u cos z ; z u sin z=il

7 Cohen, Rodriguez Villegas, and Zagier: Convergence Acceleration of Alternating Series 9 i CB i il D(1) D(r ) i CA i il i i FIGURE 1. The sets CA and CB. where L = log( p +1) (or, more precisely, any value log( p + 1) + in). To nd the limiting value of ja n ( 1)j 1=n, we need to look at the values of u for which the expression on the right becomes singular. This clearly happens if u cos z = 1 for a value of z with z (u=) sin z = il: The smallest value of juj occurs for z = i A = and equals 1= A, with A and A as in the proposition. Hence lim sup n!1 ja n ( 1)j 1=n = A. A more careful analysis gives A n ( 1) n+1= A = p n( A 1) or even an asymptotic expansion, but we will not use this. Now if we used the proof of Proposition 1 directly the error term R A n would be estimated essentially by M n =A n ( 1), where M n = max ja n (x)j: x1 Using (11), one can show that this number grows like (1:588 : : :) n (the maximum is attained at x = 1 if n is even), leading to an error estimate of about 5:3 n, which is worse than we had before. But of course the motivation for introducting the polynomials P n (m) and the diagonal subsequence A n was to improve the error term by assuming that the function w(x) or h(t) was smooth and use repeated integration by parts; see (6). Making this assumption and doing the integration by parts, we obtain A n ( 1) R A n = = = 1 n n! = ( 1)n n n! so by Taylor's theorem n= A n ( 1) R A n un = A n (x) w(x) 1 + x dx A n (sin t) h(t) dt d n dt n sin n (t) h(t) dt sin n (t) dn h(t) dt n dt; h t u sin t dt: As t goes from to = for a xed (complex) value of u, the argument t u sin t moves along a path C u connecting the points and =. If for some r S> the function h(t) is entire in the region D(r) = C jujr u, then the above integral representation of P An ( 1) Rn A un shows that this sum is analytic in the disc juj < r and hence that lim sup ja n ( 1)R A n j1=n 1=r: n!1 The best value of r we can hope for (unless w is very special) is the number r = :736 : : :, for which the point t = il lies on the boundary of D(r), since at this point the denominator 1 + sin t of h(t) will vanish. (The value of r can be computed by simple calculus: r is the minimum p of p t + L =jsin tj for t [; =] and is equal to 4t = sin 4t with t as

8 1 Experimental Mathematics, Vol. 9 (), No. 1 in the proposition.) This then leads to the estimate lim sup n!1 jrn A j 1=n 1 A with A = r A, and is valid whenever h is analytic in D(r ) or equivalently, whenever w is analytic in the region fsin t j t D(r )g, which is the region bounded by Curve CB in Figure 1. If w(x) has a singularity within this region then we get a worse estimate. In particular, if w(x) has a singularity at the origin, as in the examples 3{6 above and many others, then the convergence of Sn A to S will be like n A if h(t) has no singularities in D(1), since r = 1 is the largest value of r for which is not in the interior of D(r). The boundary of the region fsin t j t D(1)g is shown as Curve CA in Figure 1. The case of Algorithm B is very similar, but slightly more complicated. We sketch it briey. To be able to apply Lagrange's formula (1), we introduce C m (t) = 1 m! m d dt so that m (e it sin t) m e it sin t B m (sin t) = C m+1(t) + C m+1 ( t) im for m > : Then a similar use of Lagrange's formula (1) shows that lim sup jc m (il)j 1=m = B; m!1 so that lim sup jb n ( 1)j 1=n = B when n! 1 through even values of n. A similar proof shows that the same holds for odd values. The rest of the proof is essentially unchanged, still using the functions C m (t). Since e it is of modulus 1 when t is real, the domains of analycity required for w(sin t) are still the same. We thus obtain lim sup m!1 jb m ( 1)Rmj B 1=m 1=r and similarly for m+1, hence lim sup n!1 jb n ( 1)Rn B j 1=n 1= p r as claimed. Remarks. 1. More generally we can consider Algorithm Q with Q n proportional to P n (m) with m = n=(1 + ) + O(1) and >, Algorithm A corresponding to = and Algorithm B to = 1 (the exact choice of m has no eect on the speed of convergence of the algorithm). The same analysis as before shows that the analogue of Proposition remains true (with the same curves CA and CB as before), but with the numbers and occurring there replaced by = e (cosh + sinh ) 1=(+1) ; = r 1=(+1) ; where = is a root of 1=(coth + ) = L and r = :736 : : : is the same number as before. It can be shown that =, i.e. Algorithm A, gives the largest value of, and that = 1, i.e. Algorithm B, gives the largest value of, whence the choice of these two algorithms. (Note: the largest value of is in fact obtained for = :13559 : : :, but we must restrict to non-negative values of since otherwise more than n terms of the sequence fa k g are used.). We do not claim that the sequences of polynomials that we have given give the best results, only that they are natural choices. Other sequences of polynomials P n can be used for linear acceleration of alternating sequences which for certain classes of sequences fa k g will give even better convergence. These sequences of polynomials are related to polynomials which are used in Diophantine approximation to get good irrationality measures for numbers such as log, or (3), following the works of Apery, Beukers, Rhin et al. 3. For sequences fa k g for which w(x) is analytic in an even larger region than the one bounded by Curve CB, we can improve the 17:93 n bound for Algorithm A by taking a linear combination of a k and some standard sequence like a k = 1 k + 1 to force w(x) to vanish at 1. Then S n = S + w( 1)(S n S ) + " n ; where the error term " n tends to faster than before (and even more than exponentially if w(x) is entire, since the modied series X Qn ( 1)R n u n = (entire in t, u) dt has innite radius of convergence), and using this with two dierent values of n to eliminate the w( 1) term we get improved approximations to S.

9 Cohen, Rodriguez Villegas, and Zagier: Convergence Acceleration of Alternating Series 11 APPLICABILITY Since we have given three algorithms, some advice is necessary to be able to choose between the three. If the sequence fa k g is very easy to compute and not too many decimal digits are required (say at most 1), we suggest using Algorithm 1, which has the advantage of being by far the simplest to implement and which does not require any storage. This is the default choice made in the Pari system for example. If the sequence fa k g already converges to at a geometric rate, then w(x) cannot be analytic and hence Algorithm 1 should again be chosen, taking into account Remark 8 on page 5. If the sequence fa k g is dicult to compute or if a large number of decimal digits are desired, it should be better to use Algorithms A or B. Since a priori one does not know the analytic behavior of w(x), in view of the examples which have been presented, w(x) has frequently a singularity at x =, hence we suggest using Algorithm B. Of course, if we know that w(x) is much better behaved, then Algorithm A becomes useful also. EXTENSION TO CERTAIN NONALTERNATING SERIES The algorithms given can also be used in cases where alternating series occur only indirectly. A rst example is the summation of series with positive terms. Using a trick due to Van Wijngaarden and described in [Press et al. 1988], such a series can be converted to an alternating series as follows: k=1 a k = m=1 ( 1) m b m with b m = k= k a k m : In this case the coecients b m of the alternating sum are themselves innite sums, hence are hard to compute, and so it is usually useful to use Algorithm B instead of Algorithm 1. A second example is the computation of continued fractions S = b 1 =(c 1 + b =(c + )) with b k and c k positive. The standard theory of continued fractions shows that one can rewrite S as an \alternating" sum S = b 1 q q 1 b 1b q 1 q + b 1b b 3 q q 3 ; where the q i are dened by q 1 =, q = 1 and q n = c n q n 1 + b n q n, and we can then apply one of the given algorithms to the sequence a k = Q 1jk+1 b j q k q k+1 : Note that frequently continued fractions converge geometrically (this is true for example in the case of simple continued fractions, i.e. b k = 1 for all k and a k positive integers) hence Remark 8 on page 5 must be taken into account, and Algorithm 1 should usually be preferred. REFERENCES [Borwein and Crandall ] J. Borwein and R. Crandall, In preparation. [Brezinski 198] C. Brezinski, Pade-type approximation and general orthogonal polynomials, Intern. Series of Numer. Anal. 5, Birkhauser, Basel, 198. [Eiermann 1984] M. Eiermann, \On the convergence of Pade-type approximants to analytic functions", J. Comput. Appl. Math. 1: (1984), 19{7. [Gustafson 1978/79] S.- _A. Gustafson, \Convergence acceleration on a general class of power series", Computing 1:1 (1978/79), 53{69. [Hausdor 193] F. Hausdor, \Momentprobleme fur ein endliches Intervall", Math. Z. 16 (193), {48. [Hurwitz and Courant 199] A. Hurwitz and R. Courant, Vorlesungen uber allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin, 199. [Press et al. 1988] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipes in C, nd ed., Cambridge University Press, Cambridge, [Wimp 1981] J. Wimp, Sequence transformations and their applications, Math. in Sci. and Engin. 154, Academic Press, New York, 1981.

10 1 Experimental Mathematics, Vol. 9 (), No. 1 Henri Cohen, Laboratoire d'algorithmique Arithmetique et Experimentale (AX), Universite Bordeaux I, 3345 Talence, France (Henri.Cohen@math.u-bordeaux.fr) Fernando Rodriguez Villegas, Department of Mathematics, University of Texas at Austin, Austin, TX 7871, USA (villegas@math.utexas.edu) Don Zagier, Max Planck Institut fur Mathematik, Gottfried Claren Strasse 6, 535 Bonn, Germany (don@mpim-bonn.mpg.de) Received December 7, 1998; accepted January 5, 1999