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Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series

Mark B. Villarino
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Abstract

A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives three more terms and a rigorous error estimate. Finally best-possible estimates of Lodge's approximations.

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Mark B. Villarino. 2005-05-25. Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series. https://arxiv.org/abs/math/0402354

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