a question about uniform convergence of a series The problem is to investigate the uniform convergence of $\sum\limits_{n=0}^\infty \frac{x^2}{x^2+n^2}.$  Any help will be appreciated.
Thanks in advance.

As a bonus, this series not only converges but there is a closed form expression for the sum. Using the method presented here, we study the integral of $$ f(z) = \frac{x^2}{x^2+z^2} \pi \cot(\pi z)$$ along a circle of radius $R$ in the complex plane where $R$ goes to infinity. This integral vanishes in the limit, so the residues at the poles of $f(z)$ must sum to zero. Let $$S = \sum_{n\ge 1} \frac{x^2}{x^2+n^2} .$$ We thus have $$ \operatorname{Res}_{z=-ix} f(z) + \operatorname{Res}_{z=+ix} f(z) + \operatorname{Res}_{z=0} f(z) + 2S = 0,$$ where we have used the fact that $$\operatorname{Res}_{z=m} \pi\cot(\pi z) = 1,$$ with $m$ an integer. 
This gives $$\frac{x^2}{-2ix} \pi\cot(-\pi ix) + \frac{x^2}{+2ix} \pi\cot(+\pi ix) + 1 + 2S=0$$ or $$ \frac{x^2}{2ix} \pi\cot(\pi ix) + \frac{x^2}{2ix} \pi\cot(+\pi ix) + 1 + 2S=0$$ or $$ \frac{x}{i} \pi\cot(\pi ix) + 1 + 2S=0$$ and finally $$ S = -\frac{1}{2} + \frac{\pi}{2}x\coth(\pi x).$$ It follows that for the original sum with the term at $n=0$ we have $$\sum_{n\ge 0} \frac{x^2}{x^2+n^2} = \frac{1}{2} + \frac{\pi}{2}x\coth(\pi x).$$
