I do not know exactly whether this summation $\displaystyle \sum_{n=1}^{\infty} |\sin(x_{n})|$, with $x_{n}$ approaches $0$, cannot be $\infty$ or not.
But I guess it is actually divergent but does not approach $\infty$ like $\lim \sin(n)$. I try to prove it does not go to $\infty$ by definition of limit but still not succeed.
Updated for my trouble:
If a limit diverges, there are 2 cases.
Case 1: the limit approaches infinity, like lim n.
Case 2: the limit is unidentified, like lim sin(n).
So, I just want to verify that that summation above does not always go to infinity. And, yes, I still cannot do it.
$\frac {\sin {|x_n|}} {|x_n|} \to 1$ so the series converges iff $\sum |x_n|$ converges. An example where it diverges is given by $x_n=\frac 1n$. [Note that $|\sin (x_n)|=\sin |x_n|$ at least for large $n$].
An example where it converges is given by $x_n=\frac 1{n^{2}}$.