On Leibniz series defined by convex functions

摘要

It is shown that for every alpha > 0, we haveSigma(k=n+1)(infinity) (-1)(k-1)/k(alpha) = 1/2(n+theta(n))(alpha)for some strictly decreasing sequence (theta(n))(ngreater than or equal to1) such that1/2 < theta(n) < 1/2 (1 + 1/2n+1)(alpha+1)hence with lim(n-->infinity) theta(n) = 1/2. This is only a particular case of more general new results on Leibniz series defined by convex functions. (C) 2004 Elsevier Inc. All rights reserved.