Convergence of the series $sum_{n = 1}^{+infty}{left(nsin{frac{1}{n}}right)^n}$

I have to study the convergence of the series

$$\sum_{n = 1}^{+\infty}{\left(n\sin{\frac{1}{n}}\right)^n}$$

and

$$\sum_{n = 1}^{+\infty}{\left(\left(n\sin{\frac{1}{n}}\right)^n - 1\right)}.$$

I know I should study the limit

$$\lim_{n\to +\infty}{\left(n\sin{\frac{1}{n}}\right)^n}$$

and that

$$\lim_{n\to +\infty}{n\sin{\frac{1}{n}}} = 1$$

but I don't see how it helps. Any ideas ?

Thank you in advance !

Answer

On the interval $(0,1)$ we have

$$1-\frac{x^2}{6} \leq \frac{\sin x}{x}\leq e^{-x^2/6}$$
hence $\left(n\sin\frac{1}{n}\right)^n$ behaves like $e^{-\frac{1}{6n}}=1-\frac{1}{6n}+O\left(\frac{1}{n^2}\right)$ for large values of $n$ and the given series are divergent.