Now make a guess: Will you get through an oral exam with this answer even when they will let you speak this long or rather not ?

The answer is simple: No.

Because in looking at divergent series, showing one partial sum of those diverge is enough. No matter how the "other" may converge when you change the field that is underlying. I would just disprove this one by shifting this series with an epsilon to the right or to the left in your denominator which completely sends this "proof" to hell by looking at local maxima.

Or by completely omitting this argument that real and complex numbers are in any way comparable when only looking at the analytical functions over these fields and as such analytical series that converge over these fields. This easily follows from set theory by using bijections, because i just gotta find one real smooth series that is not holomorphic. I add it to yours and this explanation goes POOF. And at this point i am not even bringing the usual definition https://en.wikipedia.org/wiki/Holomorphic_function s fully into play. Gotta check those https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations right ?

This task is just not well enough given.

And obviously the question was asked over real numbers in the first place. Ramanujan just played "smart ass" here once more, because in all oral exams i know this question (when its even asked) is asked more precisely by defining the underlying field.

Ramanujan surely had some fascinating ideas but honestly : Did you ever see a even a remotely applicable one ? Anyone of these with a fascinating corollary or something like that ?

For me he is just like a demigod that spoke in a language we still aren't able to understand and possibly won't be in this century.

The real shame her is: Physicists already try to use this one before its anywhere near to actually being understood.