Sum of alternating-like series as definite integrals

• Sergio A. Carrillo ^[1]
  1. [1] Universidad Sergio Arboleda

     Universidad Sergio Arboleda

     Colombia

     
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 53, Nº. 10, 2022, págs. 2856-2866
• Idioma: inglés
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• Resumen

  • 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.