To show $\sum_{n=1}^{\infty}\frac{x}{n^{\alpha}(1+nx^{2})}$ converges uniformly

Question

Show that

$\sum_{n=1}^{\infty}\frac{x}{n^{\alpha}(1+nx^{2})}$ converges uniformly on each bounded interval in $R$ provided that $\alpha>\frac{1}{2}$. Is the convergence uniform on $R$?

I am working on the above problem. Do I need to apply Weierstrass M-test here?

Thanks.

Answer 1

Let $f_n(x)=\frac{x}{n^\alpha (1+nx^2)}$. Then, using the AM-GM inequality, we see that

$$|f_n(x)|\le \frac{1}{2n^{\alpha +1/2}}$$

Since $\sum_{n=1}^\infty \frac{1}{2n^{\alpha+1/2}}$ converges for all $\alpha>1/2$, the Weierstrass M-test guarantees that $\sum_{n=1}^\infty f_n(x)$ converges uniformly for all $x\in \mathbb{R}$.