The fastest series for $1/\pi$ due to Ramanujan. Proofs from modular polynomials

TL;DR

This paper presents general formulas for proving Ramanujan series for 1/π, demonstrates their application to Ramanujan's fastest series using modular polynomials, and offers a Maple tool for automatic proofs.

Contribution

It introduces a unified approach to prove Ramanujan series for 1/π and provides a computational tool for automatic verification.

Findings

Complete proofs of Ramanujan's fastest series for 1/π using modular polynomials

General formulas applicable to real and complex Ramanujan series

A Maple program for automatic proof generation

Abstract

First we give general formulas for proving real or complex Ramanujan series for $1/\pi$. Then, as an example, we apply them for providing complete proofs of the fastest series for $1/\pi$ due to Ramanujan using Russell and Weber modular polynomials. We recommend the reader to use a Maple program which is in the web of the author for automatically proving any Ramanujan-type series for $1/\pi$.