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The Ramanujan Journal

, Volume 47, Issue 2, pp 253–265 | Cite as

Concerning an infinite series of Ramanujan related to the natural logarithm

  • David M. Bradley
Article
  • 149 Downloads

Abstract

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.

Keywords

Completely monotone function Euler’s constant Bernoulli numbers Natural logarithm 

Mathematics Subject Classification

Primary: 65D20 Secondary: 11Y60 11B68 26D15 26A48 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MaineOronoUSA

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