Ramanujan’s Easiest Formula

A while ago I wanted to figure out how to prove one of Ramanujan’s formulas. I figure this is the sort of thing every mathematician should think about 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 crac...