Let $$f(x,y)=\sum_{n=1}^\infty \frac{x}{x^2+yn^2}, \ \ \ \ y\neq 0$$ how to show that for each $y\neq 0, g(y)=\lim_{x\rightarrow \infty} f(x,y)$ exists, evaluate $g(y)$. Then prove the convergence above if uniformly for $y>0$ as $x\rightarrow +\infty$.
For each x and y, the series converges, but I don't know how to get the sum of the series.
Hint: The series has a well-known closed form in terms of elementary functions that is convenient for finding the limit as $x\rightarrow\infty$:
$$\sum_{n=1}^\infty \frac{x}{x^2+yn^2}=\frac{\pi}{2\sqrt{y}}\coth{\left(\frac{\pi x}{\sqrt{y}}\right)}-\frac{1}{2x}.$$
A proof can be found in this question.