<s>[INST] Check for convergence $$\sum_{n = 2}^\infty (-1)^n \sin\left( \frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n  (\ln n)^2}\right)$$
I tried to use Maclaurin series, but failed to evaluate little-o.
 [/INST] Write $\sin x=x-x^3/6+x^3\varepsilon(x)$, where $\varepsilon$ is such that $\lim_{x\to 0}\varepsilon(x)=0$. Consequently, 
$$ (-1)^n \sin\left( \frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n  (\ln n)^2}\right)=(-1)^n\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n  (\ln n)^2}\right)-\frac{(-1)^n}6\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n  (\ln n)^2}\right)^3\\
+\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n  (\ln n)^2}\right)^3\delta_n,$$
where the sequence $\left(\delta_n\right)_{n\geqslant 1}$ converges to $0$ as $n$ goes to infinity. Since 
$$\left|\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n  (\ln n)^2}\right)^3\right|\leqslant cn^{-3/2}$$
for some $c$ independent of $n$ and the series $\sum_{n\geqslant 2}1/(n(\ln n)^2)$ is convergent, the problem reduces to the simpler one: determine the convergence of the series $\sum_{n\geqslant 2}(-1)^n\sin(3n)/\sqrt n$.
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