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Concerning an infinite series of Ramanujan related to the natural logarithm

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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.

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  1. Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, ME, 04469-5752, USA

    David M. Bradley

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  1. David M. Bradley
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Bradley, D.M. Concerning an infinite series of Ramanujan related to the natural logarithm. Ramanujan J 47, 253–265 (2018). https://doi.org/10.1007/s11139-017-9961-y

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  • DOI: https://doi.org/10.1007/s11139-017-9961-y

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