Learning from Ramanujan: Elementary Approaches to Profound Ideas

TL;DR

This paper provides accessible proofs of Ramanujan's results using elementary methods, highlighting their interconnectedness and relevance to modern mathematics.

Contribution

It offers new, simplified proofs of Ramanujan's identities employing classical tools, enhancing understanding and exploring extensions in number theory and analysis.

Findings

Reproved and extended Ramanujan's identities using elementary methods

Demonstrated the interconnectedness of Ramanujan's notebooks

Highlighted the relevance of Ramanujan's methods in modern mathematics

Abstract

We revisit several entries from Ramanujan's notebooks which follow from more elementary arguments than a first glance may suggest. Our goal is to demystify these results through more accessible proofs, while also shining some light on the web of interconnections within the notebooks and demonstrating the continuing relevance of Ramanujan's methods. Classical and modern tools, such as multisection, telescoping sums, partial fraction decomposition and Fourier analysis, are employed to reprove and extend identities originally presented without explanation. These contributions try not only to enrich our understanding of Ramanujan's intuition but also to offer new avenues for exploration in number theory, special functions and mathematical analysis.