<s>[INST] 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
 [/INST] By the Taylor series expansion, as $u \to 0$, one has
$$
\sin u=u+o(u^2)
$$ giving, as $n \to \infty$,
$$
\sin\left(\frac{1}{\sqrt{n+2}}\right)\times \frac{n}{(n^2+3)}=\frac{1}{\sqrt{n+2}}\times \frac{n}{(n^2+3)}+O\left( \frac{1}{n+2}\times \frac{n}{(n^2+3)}\right)
$$ or
$$
\sin\left(\frac{1}{\sqrt{n+2}}\right)\times \frac{n}{(n^2+3)}=\frac{1}{n^{3/2}}+O\left( \frac{1}{n^2}\right)
$$ and the initial series is convergent.
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