Ramanujan's formula for the harmonic number

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.