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, allows one to calculate Sigma(-1)(k)a(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.