Limit of a peculiar sum $\sum_{n=1}^{\infty}\frac{x}{1+x^2n^2}$ Find the limit (is exists) of $$\lim_{x\longrightarrow 0^+}\sum_{n=1}^{\infty}\frac{x}{1+x^2n^2}.$$
I tried turning it into Riemann sum, but to no avail as of yet.

You can actually intepret it as a Riemann sum in the following way
$$\lim_{x\rightarrow 0+}\sum_{n=1}^{\infty}\frac{x}{1+(xn)^{2}}=\lim_{m\rightarrow\infty}\sum_{n=1}^{\infty}\frac{1}{m}\frac{1}{1+(\frac{n}{m})^{2}}=\int_{0}^{\infty}\frac{dt}{1+t^{2}}=\frac{\pi}{2}$$
