The Ramanujan Journal

, Volume 46, Issue 2, pp 373–387 | Cite as

Ramanujan-like series for \(\frac{1}{\pi }\) involving harmonic numbers

  • John M. Campbell


We introduce new classes of Ramanujan-like series for \(\frac{1}{\pi }\), by devising methods for evaluating harmonic sums involving squared central binomial coefficients, such as the Ramanujan-type series
$$\begin{aligned} \sum _{n=1}^{\infty } \frac{\left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 \left( H_n^2+H_n^{(2)}\right) }{16^n (2 n-1)} = \frac{4 \pi }{3}-\frac{32 \ln ^2(2) - 32 \ln (2) + 16 }{\pi } \end{aligned}$$
introduced in this article. While the main technique used in this article is based on the evaluation of a parameter derivative of a beta-type integral, we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for \(\frac{1}{\pi }\) containing harmonic numbers.


Ramanujan-like series Harmonic number Pi formula Complete elliptic integral 

Mathematics Subject Classification

Primary 33C75 33C20 Secondary 65B10 



The author would like to thank Dr. Jonathan Sondow for a useful discussion concerning Ramanujan-like formulas for \(\frac{1}{\pi }\). The author would also like to thank two anonymous reviewers for many useful comments.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.York UniversityTorontoCanada

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