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:
• Ramanujan’s easiest formula.
Here is Ramanjuan’s puzzle in the The Journal of the Indian Mathematical Society:
Azimuth 
“I have discovered a truly marvelous proof of this formula, which, however, this server is not large enough to store.” :-)
The title made me think of Najunamar’s theorem from “Gödel, Escher, Bach”: every even prime is the sum of two odd numbers.
This could be a mathologer video
Note that in the post you mention a paper by Cauchy, while in the slides you correctly refer to Jacobi
Whoops! I’ll fix that.
That proof was really satisfying. Almost makes me miss calculus.
Thanks! Calculus is still there, waiting for you. It misses you too.