# Convergence/Divergence of infinite series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$

It is well known that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon}}$ is convergent for a fixed positive value of $\epsilon$.

It is not difficult to show that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$ is divergent using Limit comparison test with $\displaystyle\frac{1}{n}$. There is a post on this question here.

Now comes my questions:

(i) Is $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$ convergent or divergent? (I have tried several tests, like: comparison/limit comparison tests, but fail to get conclusion. My intuition is that it is divergent...)

(ii) It was stated here that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2-\cos n}}=\sum_{n=1}^{\infty} \frac{1}{n^{1+(1-\cos n)}}$ is divergent. So is there is general way to determine if $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+f(n)}}$ with $f(n)>0$ for all natural number $n$, a convergent or divergent series?