Normal convergence versus uniform convergence

Question

I am wondering about the nature of uniform and normal convergence. I know that uniform convergence is a weaker condition than normal convergence and that normal convergence even implies uniform convergence, but there is a situation that has me confused. My question is as follows:

If $f(z)$ is some complex valued function of the form $$ f(z)=\sum\limits_{n=1}^\infty f_n(z) $$ for some partial sums $f_n(z)$ so that $f(z)$ is uniformly convergent on any compact set $G \subset \mathbb{C}$, does it follow that $f(z)$ is normally convergent on $\mathbb{C}$? Or does a problem arise from the fact that we are convergent only on bounded subsets of $\mathbb{C}$, and so can say nothing about unbounded sets? Thank for any help in advance!

Answer 1

If we consider following simple example,

$$f_n(z)=-\frac{1}{n}$$ for all $z\in \mathbb C$.

Note that $\sum f_n$ converges uniformly as its just convergent alternating series of numbers but it doesn’t converge normally.

Above example should also clear your thoughts about role of bounded-unbounded subsets.