How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets

Question

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind completed) )

Answer 1

Ramanujan's reasoning is analysed in detail by Berndt, in his discussion of chapter 6 of Ramanujan's notebooks. Ramanujan starts from the Euler-MacLaurin summation formula, $$\sum_{k=1}^x f(k)=C +\int_1^x f(t)\,dt+\tfrac{1}{2}f(x)+\sum_{k\geq 1}\frac{B_{2k}}{(2k)!}\frac{d^{2k-1}}{dx^{2k-1}}f(x),$$ and focuses on the constant $C$, which he uses to find the asymptotic expansion of $\sum_{k=1}^x f(k)$ as $x\rightarrow\infty$. This approach was made somewhat more rigorous by Hardy.

In this way Ramanujan obtained the notorious identities $\sum 1 =-\tfrac{1}{2}$ and $\sum k = -\tfrac{1}{12}$. For a $p$-adic interpretation of the latter sum, see this MO post.