# Whether the critical point of a meromorphic function lies in domain of convergence

Consider a meromorephic function $$V(z)=\frac{1}{z}+\sum_{l=0}^{\infty}\kappa_{l+1}z^{l}$$, where $$\kappa_{l}\in \mathbb{R}$$ for all $$l$$'s. Then such function has a domain of convergence near 0. Suppose one can also analytic continuate it to certain bigger domain (still denoted as $$V(z)$$), and in particular, to the whole $$\mathbb{R}_{\ge 0}$$, where $$V(z)$$ is analytic on $$\mathbb{R}_{>0}$$, and $$V^{'}(z_{c})=0$$ for some $$z_{c}>0$$.

The question is do we have that $$z_{c}$$ lies in the domain of convergence of $$V(z)$$ at 0, i.e, is $$\sum_{l=0}^{\infty}\kappa_{l+1}z_{c}^{l}$$ a convergent series? If $$\kappa_{l}$$'s are all nonnegative, then certainly we see a critical point before the function goes to infinity, but I'm not sure what if there are infinite many $$\kappa_{l}$$'s that are negative. If this doesn't neccesarily hold, what other constriants we can add to make sure that it happens?

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