It's been more than a year since I took a course in which we study sequences and series of functions.
In some notes I read (without proof) that if a sequence of functions $f_n$ doesn't converges uniformly (in some interval), then the series of functions $\sum f_n$ so does neither. This might be just a trivial fact that I am not seeing.
I mean, for regular series $\sum a_n$ we always need $\lim a_n=0$ or the series will diverges. Similarly, we need $\lim |f_n-f|_A=0$ to the sequence of functions $f_n$ to converges uniformly to $f$ in $A$, so it seems reasonable that this be a necessary condition to the series $\sum f_n$ to converge uniformly.
So, I am asking, can this be use as a criterion of non-uniform convergence of $\sum f_n$? How can we prove this?