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

TL;DR

This paper provides a complete proof of Ramanujan's famous rapidly converging series for 1/π, utilizing hypergeometric series, lattice sums, and algebraic number theory techniques.

Contribution

It offers a rigorous proof of Ramanujan's π formula using hypergeometric functions and algebraic number theory, which was previously lacking.

Findings

Proof of Ramanujan's π series using hypergeometric series

Application of lattice sums by Zucker and Robertson

Incorporation of algebraic number theory results

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.