# convergence/divergence of $\sum \left(n\sin\left(\frac{1}n\right)\right)^n$

convergence/divergence of $\sum \left(n\sin\left(\frac{1}n\right)\right)^n$

Ratio and root test both are inconclusive (makes sense since this is after i found the ratio of a power series). i tried some things but it always comes down to $1^\infty$

any tips?

Hint: for large $n$s, $n\sin\frac{1}{n}$ behaves like $1-\frac{1}{6n^2}$, and $\left(1-\frac{1}{6n^2}\right)^n$ is $1-O\left(\frac{1}{n}\right)$, hence the general term is not infinitesimal and the series cannot be convergent.