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

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: $$\frac{1}{\pi }=\frac{2\sqrt{2}}{9801}\sum _{n=0}^{\mathrm{\infty }}\frac{26390n+1103}{{(n!)}^4}\frac{\left(4n\right)!}{{396}^{4n}}$$ 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.