Need help with a (apparently) difficult series

[Sebacide]

Homework Statement

I have a series, but i cannot find a method to study the convergence. Can anyone, please, help me with this series? I can't understand what kind of method can be used to study its convergence.

Relevant Equations

I thought about using ##0\leq\left|\sin(n)\right|\leq1##

This is the series: $$\sum_{n=1}^{+\infty}\sin(n)\sin\left(\frac{1}{n}\right)\left(\cos\left(\frac{1}{\sqrt{n}}\right)-1\right)$$

 

[FactChecker]

Good start. I'm sure that you will need to use the bound on ##\sin(n)## that you mention, but you need more than that. I think it must have something to do with the small angle approximations. For large ##n##, ##\sin(1/n) \approx 1/n## and ##\cos(1/\sqrt{n}) \approx 1- \frac{1}{2n}##. These approximations can be derived from the Taylor series expansions. But I have concerns about using approximations in an infinite series since I do not know how fast the terms approach their approximate value.