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Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem

Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$. Q: Suppose for some reason (e.g. numerical) we know that there is ...
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