Series of meromorphic functions meromorphic?

Question

Let $U \subseteq \mathbb C$ be open with $0\in U$. For each $m \in \mathbb N$ we have a holomorphic function $f_m:U \setminus\{0\} \to \mathbb C$ with simple pole at $s=0$. We know that $$F(s) := \sum_{m=1}^\infty f_m(s)$$ converges locally normally for $s \in U$ with $\Re(s)>0$ to a holomorphic function. Further, we know that $$a_{-1}:= \sum_{m=1}^\infty \operatorname{res}_{s=0}(f_m(s))$$ converges. Does this imply that $F$ has a meromorphic continuation in a neighborhood of $s=0$ with simple pole at $s=0$ with residue $a_{-1}$? Further, if this is the case, is Laurent series at $s=0$ of $F$ simply the sum of the Laurent series of the $f_m$?

Answer 1

No. In fact there are various ways to show there exist polynomials $p_n$ with $$\sum p_n(s)=e^{1/s}\quad(\Re s>0).$$