Resumen de Sum of alternating-like series as definite integrals

Sergio A. Carrillo

• The aim of this note is to sum the series of typean/(λn + 1), where λ > 0, the sequence (an)n≥ 0 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.