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$z \in \mathbb{C} \setminus \mathbb{Z}$

$f(z)=\sum_{k=1}^{\infty}\frac{1}{(z-k)^2}+\sum_{k=1}^{\infty}\frac{1}{(z+k)^2}+\frac{1}{z^2}$.

I want to show that $f$ is holomorphic.

I proved that the second series converges uniformly and that the first series converges uniformly on $K:=\{w \in \mathbb{C}: |w| \ge n\}\setminus \mathbb{N}$.

What do I do now? Am I finished according to Weierstrass lemma or is there more to prove?

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    $\begingroup$ On an open disc, if you discard a finite number of summands, the remaining summands will form a uniformly convergent series, so will converge to a holomorphic function on the disc. Restoring the discarded summands gives a meromorphic function there. $\endgroup$ Commented Jun 9, 2018 at 12:28
  • $\begingroup$ The notation $\{w \in \mathbb{C}: |w| \ge n\}\setminus \mathbb{N}$ doesn't make much sense to me... $\endgroup$ Commented Jun 9, 2018 at 15:00

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