The following talk is significantly too long. Some parenthetical remarks are easy enough to excise, but what else should I drop?
The talk is available here (pdf). I will give it on Thursday at "Many Cheerful Facts", a brown-bag student-organized talk series in which different graduate students present general-audience material: if you know the subject already, you won't learn anything in the talk. Someone bakes something tasty each week.
Abstract: Whereas modern physicists write down divergent series all the time, mathematicians through the ages have been variously terrified or only mildly scared of such sums. In this talk, I will survey the most important methods of summing divergent series, and make general vague remarks about them. I will quote many results, but will studiously avoid proving anything.