Integral estimates for convergent positive series

It is shown that for every α>1, we have ∑k=n+1∞1kα=1(α−1)(n+θn)α−1 for some strictly decreasing sequence (θn)n⩾1 such that 12<θn<14[1+(1+12n+1)α], hence with limn→∞θn=12. This is only a particular case of more general new results on series defined by convex functions.