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I am given that $f: D(P, r) \setminus \{ P \} \to \mathbb{C}$ is holomorphic. I want to show that the principal part of the Laurent series expansion for $f$ converges absolutely on $\mathbb{C} \setminus \{P\}$. Confused on how to proceed.

I know that the convergence is absolute on $D(P, r) \setminus \{ P \}$. I don't see how to extend the principal part to $\mathbb{C}\setminus\{P\}$.

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The principal part gets better as you get farther from the pole, since it has only negative powers of $(z - P)$. Apply a comparison test, using the fact that the series is convergent at some point $z \ne P$.

In particular, you can use the fact that

$$\left|\frac{a_n}{(z - P)^n}\right| \le \frac{|a_n|}{r^n}$$

for $z \notin \mathbb{D}(P, r)$.

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  • $\begingroup$ Thank you. I will try this out. $\endgroup$ – user623740 Dec 7 '18 at 0:36
  • $\begingroup$ So basically take the limit of the right-hand side as $r \to \infty$? $\endgroup$ – user623740 Dec 7 '18 at 1:33
  • $\begingroup$ No, because $r$ is just some fixed number. $\endgroup$ – user296602 Dec 7 '18 at 1:34

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