Let $(f_k)_{k \in \mathbb{N}_0}$ be a sequence in $\mathbb{R}$ s.t. $\lim_{k \to \infty}f_k =0$. Consider $$ F(z):=\sum_{k\geq 0} z^{k}f_k.$$ Of course $F$ is holomorphic on the unit disk. If $f_n =1\forall n$ one would have a meromorphic extension of $F$ to $\mathbb{C}$ with a simple pole at $z=1$.
My question now is: Is it still true that $F$ can be meromorphically extended to some domain containing $\{z:\|z\|<1+\epsilon\}$ for some $\epsilon>0$ s.t. it has at most a pole at $z=1$ ? (More generally: Is there anything one can say about meromorphic extensions of $F$ ?) I am thankful for any suggestion...
I know that the answer to the question is negative in general if $f_n$ is only bounded instead of convergent to $0$. Do you have any ideas?