## Generating series

Last Wednesday morning lecture, to calculus students, in two sentences:

A series that looks like $\displaystyle\sum_{n = 0}^\infty a_n x^n$ is called a Taylor series.  We are very interested in understanding these series and the functions they represent, so it is of the utmost importance that we pay careful attention to the conditions under which they converge.

Last Wednesday afternoon lecture, to combinatorics students, in two sentences:

A series that looks like $\displaystyle\sum_{n = 0}^\infty a_n x^n$ is called a generating function.  We are very interested in understanding these series and the sequences they represent, so it is of the utmost importance that we carefully ignore all issues of convergence.

My colleague JSK offers reassurance: “It’s easy, as long as you keep in mind that Taylor series are functions and generating functions are series.”