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Inspired by this question, once we know the radius of convergence for a formal power series...

Is there some systematic method to find out (if it is a meromorphic function), where the pole is situated that limits it's radius of convergence?

We can take the function series from the previous question as an example: $$f(x) = x\sqrt1+x^2\sqrt2+x^3\sqrt3+\cdots+x^n\sqrt{n}+\cdots$$

  1. How to decide if it corresponds to a meromorphic function.
  2. If that is the case, how to decide which pole is blocking our convergence.
mathreadler
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  • Since the coefficients are real nonnegative, the pole closest to the origin is obviously $x=1$. – Did Sep 02 '17 at 09:40
  • Yes, but how in general. – mathreadler Sep 02 '17 at 09:42
  • Are you asking for a general way to determine the points of the circle ${z;|z|=R}$, when $R$ is positive and finite, that are poles of a given series $f(z)$? There are none. – Did Sep 02 '17 at 09:45
  • I suppose that is what i am asking. Is there some result showing that there can't be any or just that none have been found so far? – mathreadler Sep 02 '17 at 09:48

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