Bruce Berndt has a series of books I think proving as much as possible from "Ramanujan's Notebooks" (which are their titles: I, II, III). Here is a summary article he wrote about these books: <https://faculty.math.illinois.edu/~berndt/articles/aachen.pdf>. —Dan Cris Moore wrote: ----- This is a beautiful article, but it doesn’t explain where any of these identities (Ramanujan and successors for \pi) come from. I know this might be a heavy lift, but are there any of these that can be explained on an elementary level, even if non-rigorously? -----