Uniform convergence of the series

Question

Test the uniform convergence of the series

$$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$

$$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$

Can I find $M_n$ such that $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}\lt \sum_{n=1}^{+\infty}M_n<+\infty\;\;\;\;\;\;\;?$$

Answer 1

Actually, if $|z2−n2π2|≥n2π2$, then your initial inequality holds.

Answer 2

You just need a sequence $M_{n}$ such that $$|\frac{1}{z^{2} - n^{2}\pi^{2}}| \le M_{n}$$ and se you can take $M_{n} = \frac{1}{n^{2}\pi^{2}}.$

EDIT: Wait, what I wrote is just nonsense. The bound goes the other way in my example. Please ignore it.