On the elegance of Ramanujan's series for $\pi$

TL;DR

This paper revisits Ramanujan's elegant series for 1/π, providing a traditional proof and exploring the mathematical identities involved, highlighting their beauty and complexity.

Contribution

It offers a clear presentation of Ramanujan's series for 1/π along with derivations involving classical and modular functions, emphasizing their mathematical elegance.

Findings

Re-presentation of Ramanujan's series for 1/π

Derivation involving classical and modular functions

Illustration of the series' mathematical elegance

Abstract

Re presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of $\pi$ :\begin{equation}\frac{1}{\pi} = \frac{\sqrt{8}}{9801} \sum_{n=0}^{+\infty} \frac{(4n)!}{(n!)^4} \frac{1103 + 26390n}{396^{4n}} \; \text{,}\end{equation}as well as several other examples of Ramanujan's infinite series. As a matter of fact, the derivation of such formulae has involved specialized knowledge of identities of classical functions and modular functions.