Sum of alternating-like series as definite integrals

Abstract

The aim of this note is to sum the series of type , where , the sequence repeats k-periodically and the sum of its first k terms vanishes. Using elementary methods, we can transform this series to a definite integral, suitable for students of Calculus to work with. We include several examples and discuss how to write the integrals involved in terms of the digamma function. If λ is a rational, the sum can be evaluated explicitly. We give a proof of the explicit value of Dirichlet L-series as an application at s = 1.