TITLE: Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
QUESTION [17 upvotes]: 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.

REPLY [3 votes]: There is a very interesting paper by Bernard Brighi on the divergence of the series 
$\sum\limits_{n=1}^{\infty} \frac{1}{n^{2+ \cos{(a+n)}}}$ for any real $a$ which I would like to attach but I don't know how to do it. Maybe most people who like the series already know it
The link is here: 
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=361997&p=1983209#p1983209