hard time with series convergence or divergence

Question

I'm having real hard time with this series

I can't prove that the series converges and also I can't prove that the series diverges:

$$\sum_{k=1}^\infty\frac{\sin^2(n)}{n}.$$

any help would be appreciated.

Answer 1

An idea:

$$\sin^2n=\frac12(1-\cos 2n)$$

Now, using Dirichlet's test we get that

$$\sum_{n=1}^\infty\frac{\cos2n}n\;\;\text{converges}$$

and since the harmonic series diverges then our series diverges as well.