Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$

We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges.

One can then ask the question of testing the convergence the following 2 Series:

$$\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}, \quad \sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \sin{n}}}$$ where $ k \in (0,2)$.

Only thing which i have as tool for this problem is the inequality $| \sin{n} | \leq 1$, which i am not sure whether would applicable or not.