Ramanujan’s formula for the harmonic number

Chao-Ping Chen

Applied Mathematics and Computation, 2018, vol. 317, issue C, 121-128

Abstract: In this paper, we investigate certain asymptotic series used by Hirschhorn to prove an asymptotic expansion of Ramanujan for the nth harmonic number. We give a general form of these series with a recursive formula for its coefficients. By using the result obtained, we present a formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. We also give a recurrence relation for determining the coefficients aj(r) such that Hn:=∑k=1n1k∼12ln(2m)+γ+112m(∑j=0∞aj(r)mj)1/ras n → ∞, where m=n(n+1)/2 is the nth triangular number and γ is the Euler–Mascheroni constant. In particular, for r=1, we obtain Ramanujan’s expansion for the harmonic number.

Keywords: Harmonic number; Euler–Mascheroni constant; Asymptotic expansion (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:317:y:2018:i:c:p:121-128

DOI: 10.1016/j.amc.2017.08.053

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