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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?

Any comment or answer?

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If you replace cosine with sine, the answer is here:… – Byron Schmuland Feb 22 '13 at 1:06
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions... – pipi Feb 23 '13 at 2:25

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