Learning from Ramanujan: Elementary Approaches to Profound Ideas

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.