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Apr 19, 2013 at 7:31 comment added Loïc Teyssier I was wondering if there are still counter-examples when the power series is assumed to be convergent for all complex values of $x\neq 0$. Using Cauchy's formula it appears at first that some control exists on $\alpha_n(0)$, although there is a swap in quantifiers that needs some looking carefully, which I don't have that much time to do right now...
Apr 18, 2013 at 14:06 history edited Andrew CC BY-SA 3.0
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Apr 18, 2013 at 13:48 history answered Andrew CC BY-SA 3.0