Convergence is contagious

Suppose g is function g: \Bbb C\to \Bbb C and that at a point z_0\in \Bbb C such function is analytic. If the infinite sum g(z_0)+g'(z_0)+\cdots converges, then in fact g is entire and such sum converges for any other choice of w.

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Convergence is contagious

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