A while ago I decided to figure out how to prove one of Ramanujan’s formulas. I feel this is the sort of thing every mathematician should try at least once.
I picked the easiest one I could find. Hardy called it one of the “least impressive”. Still, it was pretty interesting: it turned out to be a puzzle within a puzzle. It has an easy outer layer which one can solve using standard ideas in calculus, and a tougher inner core which requires more cleverness. This inner core was cracked first by Laplace and then by Jacobi. Not being clever enough to do it myself, I read Jacobi’s two-page paper on this subject to figure out the trick. It was in Latin, and full of mistakes, but still one of the most fun papers I’ve ever read.
On Friday November 20th I’m giving a talk about this at the Whittier College Math Club, which is run by my former student Brandon Coya. Here are my slides:
Here is Ramanjuan’s puzzle in the The Journal of the Indian Mathematical Society: