Analiticity of function $f(z)=\sum_{n=1}^{\infty}\frac{1}{1+n^2z}$

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Please help me this problem:

Which is the largest region where $f(z)=\sum_{n=1}^{\infty}\frac{1}{1+n^2z}$ is an analitic function?, where is f a meromorphic function?

Thansk for your support.