Telescoping Estimates for Smooth Series

We derive telescoping majorants and minorants for some classes of series and give applications of these results. P ∞ k=n k −s , s > 1. The present article is dedicated to the question which se- ries can be treated in a similar way. In the following we shall show that this is the case for a big class of series P ∞=n f(k). They have to satisfy only certain mild smoothness conditions on the function f. The proofs are based on the comparison of f(k) and Z k+1−c k−c f(x) dx, c 2 (0,1), that may be regarded as special cases of theorems from the theory of numer- ical integration. We will demonstrate the usefulness of this method by some applications. Among them there will be a generalization of the Stieltjes constants and an elementary proof of Stirling's formula.