How to prove a Ramanujan-type series for Pi?

Question

I am referring to Ramanujan series or the Chudnovsky series or any similar series converging fast. When we see the formula we go WOW but is there a proof available for some series like that? Didn't Ramanujan provide a proof in his letters to Cambridge University or anywhere else?

Thanks in advance

Answer 1

Re-typeset version of Ramanujan's 1914 article Modular equations and approximations to π: http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf (number 6 from http://ramanujan.sirinudi.org/html/published_papers.html).

Lorenz Milla, A detailed proof of the Chudnovsky formula with means of basic complex analysis: https://arxiv.org/abs/1809.00533.