$f(z)=\sum_{k=1}^{\infty}\frac{1}{(z-k)^2}+\sum_{k=1}^{\infty}\frac{1}{(z+k)^2}+\frac{1}{z^2}$ holomorphic. $z \in \mathbb{C} \setminus \mathbb{Z}$

Question

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

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

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?