These notes are for a talk given in Michigan's Undergraduate Math Club.

The goal of these notes is to give a very brief introduction to divergent series, as in the first chapter of Hardy's famous book, and to show that divergent series are in fact an interesting object of study. We highlight some of the problems that arise when one tries to make sense of a divergent series without clear definitions, and this will lead naturally to two generalizations of the notion of "sum," both of which can be used to assign meaning to divergent series.