This is an account 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.