Uniform Convergence $\sum \frac{x}{n(1+nx^2)}$

Question

Analyze the uniform convergence of the following series $$\sum\limits_{n=1}^\infty \frac{x}{n(1+nx^2)}$$

My attempt- This converges absolutely and uniformly for $(-\infty,\delta]\bigcup[\delta,\infty)$ using Weierstrass Mn Test with $$M_n= \frac{1}{\delta n^2}$$

I am not able to make any conclusions for $(-\delta, \delta)$. I tried looking here and nothing weird is hapenning at $0$.

Answer 1

Using $\sqrt{n}|x|\leq \frac{1+nx^2}{2} $ we have that $$|\frac{x}{n(1+nx^2)}|\leq\frac{1}{2n^{3/2}}$$ Which by Weierstrass test gives us uniform convergence, since $\sum \frac{1}{2n^{3/2}}$ is convergent