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Let $f$ be holomorphic on an open disk containing the unit circle, except in a pole $w$ on the unit circle.

Assume that f has a power series expansion $\displaystyle \sum_{k=0}^{\infty} a_n z^n$ in the open unit disc.

Show that $\dfrac{a_n}{a_{n+1}} \to w$.

How can I do this?

crank
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  • Have you already heard of Laurent expansions? – Daniel Fischer Dec 22 '14 at 20:10
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    That series expansion doesn't seem right does it? The lower bound states $k$ and if a the function $f(z) = \sum_{n=0}^\infty a_nz^n$ then the function would be entire would it not? – dietervdf Dec 22 '14 at 20:22
  • Oh sorry I didn't asked the question properly: I have to show that the ratio converges to w, if f has a power series expansion in the open unit disc. That's indeed the same question which was asked before. Thank you – crank Dec 23 '14 at 17:01
  • I dont understand the solution completely: First of all, f has a power series expansion in the open unit disc: We know this series converges absolutely for all z inside the open unit disc and diverges for all z outside the closed unit disc, since we know that it can't converge at w, since f is not holomorphic there and hence the radius of convergence is 1. I don't understand how they argue that $a_n$ doesn't converge to 0. – crank Dec 23 '14 at 17:48

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