The summation of alternating and monotonic series

摘要

In previous papers the author has treated the question of convergence acceleration of alternating series, and of power series in x having positive coefficients. In the latter case it was assumed that 0 < x < 1, but even for x extremely close (but not equal) to 1, a powerful convergence acceleration was obtained. The present paper propounds an improved convergence acceleration method for alternating series, and also an efficient convergence acceleration technique which is valid for many monotonic series of importance in applied mathematics, i.e. the case x = 1 is now treated. In essence this technique obtains from the positive terms of the monotonic series an alternating sequence of convergents, which are then handled by the method for alternating series or sequences. It is assumed that the coefficients (or terms) of the series treated in this paper are moments of a function itf(u) over the interval 0 ⩽ u ⩽ 1, and the methods are based on properties of the shifted Legendre polynomials P∗n(u), and on the author's earlier summation formula for power series having moment coefficients. The function itf(u) does not need to be known.