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As the title already mentioned. If not, could you please give me a counterexample?

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  • $\begingroup$ Suppose we have $e^z$, its radius of convergence must be $\rho = \infty$. A power series centered at 0 converges uniformly in any ball $B(0,r)$ with $r \in (0, \rho)$. So, it converges uniformly in any ball with $r>0$. It's just what I think, please do correct me if I'm wrong $\endgroup$
    – Gvxfjørt
    Jun 13 '16 at 13:02

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