Possible Duplicate:
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
I am having a very hard time with the following series: $$\sum_{n>0}\frac{1}{n^{1+|\sin n|}}$$ I don't even have clues for guessing its character. Does it converge or not?
Of course the heart of the problem is the structure of the set
$$X=\{|\sin n|: \, n\in\mathbb{N}\}.$$
In fact $x=0$ is not in the limit class of $X$, otherwise we'd be able to find a convergent subsequence of $|\sin n|$ (and so the series would diverge); however zero is an accumulation point of $X$, so it is hopeless to try to compare our series with
$$\sum_{n>0}\frac{1}{n^{1+\varepsilon}}$$
for any fixed ε. Any ideas?
