If you sum an expression over an uncountable set $\sum_{x\in \mathbb{R}}f(x)$, then do we need $f(x)=0$ on all but a countable subset in order for the sum to have a finite value?
If not can you give an example of a function everywhere nonzero that has a transfinite sum with a finite value?
Possible keywords: Transseries, Écalle–Borel Summation, analyzable function
Transseries for beginners, GA Edgar, 2009