I while ago I posted a short note on integer partitions and generating functions. I am adding a link here, as it is obviously connected to the combinatorial applications of power series we have been considering.
There are some excellent books on the topic of generating functions. The classic, generatingfunctionology, can be downloaded for free at the website of the author, Herbert Wilf.
Another nice reference is Analytic combinatorics, by Philippe Flajolet and Robert Sedgewick. It can be downloaded for free at Flajolet’s website.
Finally, a classic in analysis that in particular covers this material nicely is the book by Pólya and Problems and Theorems in Analysis. It is a work in two volumes. Series are studied in the first one.