How to prove that $\sum_{n=1}^\infty \frac{x}{n(1+nx^2)}$ converges uniformly?

Question

I know that $$ \frac{x}{n(1+nx^2)} \leq \frac{1}{2}\frac{1}{n\sqrt{n}}, $$ but I run out of my way to prove that $$ \sum_{n=1}^\infty \frac{1}{2}\frac{1}{n\sqrt{n}} $$ converges. Please help.

Answer 1

Recall that the series $\displaystyle{\sum_{n=1}^\infty \frac{1}{2}\frac{1}{n\sqrt{n}} }$ it is a $p$-series: $$\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n^{p}}}$$ with $p=\frac{3}{2}>1$ and therefore it is convergent. Also, since $n\geq 0$ we have: $$\frac{x}{n(1+nx^2)}=\frac{x}{n+n^{2}x^2}\leq \frac{x}{n^{2}x^{2}}=\frac{1}{xn^{2}}$$ this impies by Weierstrass M-test that the series $ \displaystyle{\sum_{n=1}^\infty\frac{x}{n(1+nx^2)}}$ is uniformly convergent when $x\geq 1$.