Integral estimates for convergent positive

It is shown that for every α >1, we have ∞ k=n+1 1 kα = 1 (α − 1)(n +θn)α−1 for some strictly decreasing sequence (θn)n 1 such that 1 2 < θn < 1 4 1 + 1 + 1 2n+ 1 α , hence with limn→∞θn = 12 . This is only a particular case of more general new results on series defined by convex functions.  2004 Elsevier Inc. All rights reserved.