convergence of $\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p$

I want to find all $p\in \mathbb{R}$ such that the following series converges:$$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p.$$We know $$\frac{1}{n}-\sin\frac{1}{n}\leq \frac{1}{n^3}.$$Accordingly, for $p>\frac{1}{3}$,$$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p\leq \frac{1}{n^{3p}},$$so the series converges.

Now, I was wondering how to prove divergence of the series for $p\leq\frac{1}{3}$.