- Open Access
On the harmonic number expansion by Ramanujan
© Mortici and Chen; licensee Springer 2013
- Received: 9 December 2012
- Accepted: 12 April 2013
- Published: 3 May 2013
Let denote the Euler-Mascheroni constant, and let the sequences
The main aim of this paper is to find the values r, s, t, a, b, c and d which provide the fastest sequences and approximating the Euler-Mascheroni constant. Also, we give the upper and lower bounds for in terms of .
MSC: 11Y60, 40A05, 33B15.
- Euler-Mascheroni constant
- harmonic numbers
- psi function
- polygamma functions
- asymptotic expansion
Entry 9 of Chapter 38 of Berndt’s edition of Ramanujan’s Notebooks [, p.521] reads,
For the history and the development of Ramanujan’s formula, see .
and posed the following natural question.
is the sequence which would converge to γ in the fastest way.
Very recently, Yang  published the solution of the open problem (1.6) by using logarithmic-type Bell polynomials.
respectively. Our Theorems 1 and 2 are to find the values r, s, t, a, b, c and d which provide the fastest sequences and approximating the Euler-Mascheroni constant.
The speed of convergence of the sequence is .
The speed of convergence of the sequence is .
Our Theorems 3 and 4 establish the bounds for in terms of .
Remark 1 The inequality (1.14) is sharper than (1.8), while the inequality (1.13) is sharper than (1.14).
Before we prove the main theorems, let us give some preliminary results.
are called the polygamma functions.
Lemma 1 gives a method for measuring the speed of convergence.
Lemma 2 [, Theorem 9]
By using Lemma 1, we obtain the assertion of Theorem 1. □
By using Lemma 1, we obtain the assertion of Theorem 2. □
Therefore, for .
by using the asymptotic formula (2.3). This completes the proof of the second inequality in (1.13). □
Therefore, for .
by using the asymptotic formula (2.3). This completes the proof of the first inequality in (1.14). □
Remark 2 Some calculations in this work were performed by using the Maple software for symbolic calculations.
Remark 3 The work of the first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.
Dedicated to Professor Hari M Srivastava.
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