On Leibniz series defined by convex functions

It is shown that for every α >0, we have ∞ k=n+1 (−1)k−1 kα = 1 2(n +θn)α for some strictly decreasing sequence (θn)n 1 such that 1 2 < θn < 1 2 1 + 1 2n + 1 α+1 , hence with limn→∞θn = 12 . This is only a particular case of more general new results on Leibniz series defined by convex functions.  2004 Elsevier Inc. All rights reserved.