An invitation to formal power series

TL;DR

This paper presents a comprehensive lecture on formal power series theory, deriving key mathematical identities and theorems without using analytic methods, and exploring combinatorial applications.

Contribution

It introduces a purely algebraic approach to formal power series, proving major results like Newton's binomial theorem and Ramanujan's identities without analysis.

Findings

Proved Newton's binomial theorem using formal series

Derived Ramanujan's partition congruences algebraically

Extended methods to multivariate and Laurent series

Abstract

This is a lecture on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan's partition congruences, generating functions of Stirling numbers and Jacobi's four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon's master theorem.