Show convergence of the series of functions $\sum_{n=1}^\infty \frac{x}{n^\alpha (1+nx^2)}$ on $\mathbb{R}$, where $\alpha > \frac{1}{2}$
My attempt:
It suffices to show that for every fixed point $x \in \mathbb{R}$, the series $\sum_{n=1}^\infty \frac{x}{n^\alpha (1+nx^2)}$ of real numbers converges.
For $x = 0$, convergence is trivial.
Let $x \neq 0$. Then, we have:
$$\frac{1}{n^\alpha (1+nx²)} \leq \frac{1}{n^\alpha nx²} = \frac{1}{n^{1 + \alpha}x²}$$
and because the series $\sum\frac{1}{n^{1+ \alpha}}$ converges, the series $\sum \frac{1}{n^{1 + \alpha}x²}$ converges as well and by the comparison test the series $\sum\frac{1}{n^\alpha (1+nx²)}$ converges as well. Hence, we deduce that $\sum\frac{x}{n^\alpha (1+nx²)}$ is convergent for a fixed $x \neq 0$
Is this correct?