Study the convergence of the following series $\sum\limits_{n=1}^\infty \frac{n}{n^2+3} \sin(\frac{1}{\sqrt{n+2}}) $ I have to study the convergence of the following series:
$\sum\limits_{n=1}^\infty \frac{n}{n^2+3} \sin(\frac{1}{\sqrt{n+2}}) $
Is a positive series, so I should divide for $\frac{1}{\sqrt{n+2}}$ 
and then use the comparison test ?
Many thanks

With equivalents, as it's a series with positive terms:
\begin{align*}\frac n{n^2+3}&\sim_\infty\frac1n,\quad \sin\frac1{\sqrt{n+2}}\sim_\infty\frac1{\sqrt{n+2}}\sim_\infty\frac1{\sqrt n}, \\\text{hence}\quad\frac n{n^2+3}\sin\frac1{\sqrt{n+2}}&\sim_\infty\frac 1{n^{3/2}},\quad\text{which converges.}
\end{align*}
