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.
Categories: math.NT