Convergence Acceleration of Alternating Series

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, permits one to calculate P (?1) k a k with an error of about 17:93 ?n from the rst 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 (e.g. several hundred digits) frequently occurring in number theory. The typical series we will be considering are alternating series S = P 1 k=0 (?1) k a k , where a k is a reasonably well-behaved function of k which goes slowly to 0 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. For k = 0 up to k = n ? 1, repeat the following: c = b ? c; s = s + c a k ; b = (k + n)(k ? n)b=((k + 1=2)(k + 1)); Output: s=d. This algorithm computes an approximation to S as a weighted sum of a 0 ; : : : ; a n?1 with universal rational coeecients 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, 2, 3, 4 the approximations given by the algorithm are 2a 0 =3, (16a 0 ? 8a 1)=17, (98a 0 ? 80a 1 + 32a 2)=99, and (576a 0 ? 544a 1 + 384a 2 ?128a 3)=577, respectively. The denominator d n grows like 5:828 n and the absolute values of the coeecients c n;k decrease smoothly from d n ? 1 to 0. 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 5:828 ?n , so that to get D decimal digits it suuces to take n equal to approximately 1.31D. Notice that the number of terms and the time needed to get …