How to see that series $|\sin(1/n^2)|$ converges or diverges? How to see that series $\sum_{n=1}^{\infty}|\sin(1/n^{2})|$ converge or diverge? That is, to see if $\sum_{n}\sin(1/n^{2})$ absolutely converges.
I know that for all $n$ in $\mathbb N$, $\sin(1/n^2)<1/n^2$, but is that fact useful here? Could someone tell me how to show that?

If you're looking to prove the inequality $$\left|\sin\frac1{n^2}\right|\le\frac1{n^2},$$ it follows from:
Claim: $|\sin x|\le |x|$ for every $x$.
Proof: Suppose $x\ne0$. By the mean value theorem, there exists a number $a$ such that
$$\sin x - \sin 0 = \cos(a)(x-0),\tag1$$
since the derivative of $\sin x$ is $\cos x$. Take absolute values of both sides of (1), then use the fact that $|\cos a|\le 1$.
