Proving $\sum_{n=1}^{\infty}\frac{x^n}{n(1+nx^2)}$ converges uniformly.

Question

I want to prove the series $$\sum_{n=1}^{\infty}\frac{x^n}{n(1+nx^2)}$$ Converge uniformly in $[-1,1]$ and not in the rest of $\mathbb{R}$. I have solved similar problems to this using that a functions converges uniformly if and only if satisfies Cauchy condition, and then using the triangle inequality. In this case I can't do that since in the numerator I only have $x^n$. I have also tried to use the Weierstrass M-test without any results.

Answer 1

For $|x|>1$ the series is divergent. For the interval $[-1,1]$, try again with Weierstrass M-test.

Note that for $0<r\leq |x|\leq 1$, $$\left|\frac{x^n}{n(1+nx^2)}\right|\leq \frac{1}{n(1+nr^2)}\leq \frac{1}{r^2n^2}.$$ On the other hand, for $|x|\leq r<1$, $$\left|\frac{x^n}{n(1+nx^2)}\right|\leq r^n.$$ Therefore, by taking $r=1/2$, we have that $$\max_{x\in [-1,1]}\left|\frac{x^n}{n(1+nx^2)}\right|\leq \max\left(\frac{1}{2^n},\frac{4}{n^2}\right)=\frac{4}{n^2}.$$