Convergence acceleration of alternating series

Abstract

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 $\sum(-1)^ka_k$ with an error of about $17$.$93^{-n}$ from the first $n$ terms for a wide class of sequences $\{a_k\}$. Such methods are useful for high precision calculations frequently appearing in number theory.