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

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.