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Ramanujan’s formula for the harmonic number

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  • Chen, Chao-Ping

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.

Suggested Citation

  • Chen, Chao-Ping, 2018. "Ramanujan’s formula for the harmonic number," Applied Mathematics and Computation, Elsevier, vol. 317(C), pages 121-128.
    • Handle: RePEc:eee:apmaco:v:317:y:2018:i:c:p:121-128
    DOI: 10.1016/j.amc.2017.08.053
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    Cited by:

    1. Kwang-Wu Chen, 2022. "Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion," Mathematics, MDPI, vol. 10(12), pages 1-10, June.

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