  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.
