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arXiv · 2411.15803

A Proof of Ramanujan's Classic $\pi$ Formula

Thang Pang Ern·Devandhira Wijaya Wangsa
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Abstract

In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $\pi$. Among these, one of the most celebrated is the following series: \[\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{26390n+1103}{\left(n!\right)^4} \frac{\left(4n\right)!}{396^{4n}}\] In this paper, we give a full proof of this classic formula using hypergeometric series and a special type of lattice sums due to Zucker and Robertson. We will also use some results by Dirichlet and Edwards in algebraic number theory.

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Thang Pang Ern, Devandhira Wijaya Wangsa. 2024-11-24. A Proof of Ramanujan's Classic $\pi$ Formula. https://arxiv.org/abs/2411.15803

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