Need help with a (apparently) difficult series

• Sebacide
In summary, the conversation discusses the use of small angle approximations in dealing with an infinite series involving trigonometric functions. The speaker mentions the need for more than just the bound on ##\sin(n)## and suggests using the Taylor series expansions for the approximations of ##\sin(1/n)## and ##\cos(1/\sqrt{n})##. However, there are concerns about the convergence rate of the approximations in an infinite 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
This is the series: $$\sum_{n=1}^{+\infty}\sin(n)\sin\left(\frac{1}{n}\right)\left(\cos\left(\frac{1}{\sqrt{n}}\right)-1\right)$$

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.

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