Convergence of $\sum_n \frac{(-1)^n n^{\alpha}\sin{\frac{1}{n^{\alpha}}}}{n^{\beta} + (-1)^n}$ 
Discuss the convergence/divergence of the series of general term :$$\frac{(-1)^n n^{\alpha}\sin{\frac{1}{n^{\alpha}}}}{n^{\beta} + (-1)^n}\:\:\alpha,\beta >0$$

What I've done so far :
$$\begin{align}
\left\lvert\frac{(-1)^n n^{\alpha}\sin{\frac{1}{n^{\alpha}}}}{n^{\beta} + (-1)^n} \right\rvert \leq \left\lvert \frac{n^\alpha}{n^{\beta} + (-1)^n} \right\rvert \sim \left\lvert \frac1{n^{\beta-\alpha}}  \right\rvert  \\
\end{align}$$
which means the series converges if $ \beta > \alpha + 1 $ and it diverges otherwise.
Am I on the right path ? Did I make a mistake or miss something ?
Any clarification would be great.

Simply note that since $\alpha >0$ and $\beta >0$,
$$\frac{(-1)^n n^{\alpha}\sin{\frac{1}{n^{\alpha}}}}{n^{\beta} + (-1)^n} = \frac{(-1)^n}{n^\beta}(1+o(1))\left(1- \frac{(-1)^n}{n^\beta} +o\left( \frac{1}{n^\beta}\right)   \right)$$
hence 
$$\frac{(-1)^n n^{\alpha}\sin{\frac{1}{n^{\alpha}}}}{n^{\beta} + (-1)^n} =   \frac{(-1)^n}{n^\beta} - \frac{1}{n^{2\beta}} + o\left(\frac{1}{n^{2\beta}} \right)$$
$\displaystyle \sum_n \frac{(-1)^n}{n^\beta}$ converges no matter $\beta$.
By the limit comparison test, $\displaystyle \sum_n \left(\frac{1}{n^{2\beta}} + o\left(\frac{1}{n^{2\beta}} \right)\right)$ converges iff $\displaystyle \sum_n \frac{1}{n^{2\beta}}$ converges, that is to say $\beta > \frac 12$.
Finally, the series converges iff $\beta > \frac 12$.
