The Ramanujan Journal

, Volume 46, Issue 2, pp 373–387

# Ramanujan-like series for $$\frac{1}{\pi }$$ involving harmonic numbers

• John M. Campbell
Article

## Abstract

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.

## Keywords

Ramanujan-like series Harmonic number Pi formula Complete elliptic integral

## Mathematics Subject Classification

Primary 33C75 33C20 Secondary 65B10

## Notes

### Acknowledgements

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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