Test the series for uniform convergence

Question

$$\sum_{n = 1}^\infty \frac{nx}{1+n^2x^2}$$ How to proceed? Can't we do it using partial sum method?

Answer 1

The series diverges for any $x \ne 0$, by comparison to $\sum_{n=1}^\infty 1/n$. Note that for $x > 0$, $$ \dfrac{nx/(1+n^2 x^2)}{1/n} = \dfrac{n^2 x}{1+n^2 x^2} > \dfrac{1}{2x} \ \text{when}\ n > 1/x $$