Related:Why is this sequence uniformly convergent?
Let $K$ be a compact subset of $\mathbb{C}$.
Let $f_n$ be a sequence of continuous functions such that $f_n:K\rightarrow \mathbb{C}$.
Assume that $\sum f_n(z)$ is absolutely convergent for each $z\in K$.
Is there an example such that $\sum |f_n(z)|$ is discontinuous?
I'll give an example in $\mathbb{R}$ that I believe should work in $\mathbb{C}$. Let $f_n(x)=\frac{x^2}{(1+x^2)^n}$ on the compact set $[0,1]$. For $x=0$, obviously $f_n(x)=0$ $\forall n$ so that $\sum\limits_{n=0}^\infty f_n(x)=0$. Away from $0$, the function is a geometric series with sum $1+x^2$. Thus if $f(x)=\sum\limits_{n=0}^\infty f_n(x)$ then $f(x)$ is not continuous.