TITLE: Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？
QUESTION [64 upvotes]: Is the series
$$ \sum_{n=1}^\infty \frac{|\sin n|^n}n\tag{1}$$
convergent？
If one want to use Abel's test, is 
$$ \sum_{n=1}^\infty |\sin n|^n\tag{2}$$
convergent？
Thank you very much

REPLY [9 votes]: Note: Another great answer to this problem is given by Terry Tao in this MO post.