Ramanujan-type formulae for $1/\pi$: A second wind?

TL;DR

This paper surveys Ramanujan's 1914 series for 1/π, explores proof methods, and discusses recent generalizations, some of which remain unproven, highlighting ongoing research in this area.

Contribution

It provides a comprehensive overview of Ramanujan's formulae, proof techniques, and recent extensions, including unproven conjectures, advancing understanding of these mathematical series.

Findings

Compilation of proof methods for Ramanujan's series

Identification of new generalizations of Ramanujan's formulae

Highlighting unproven conjectures in the field

Abstract

In 1914 S. Ramanujan recorded a list of 17 series for $1/\pi$. We survey the methods of proofs of Ramanujan's formulae and indicate recently discovered generalizations, some of which are not yet proven.