TITLE: Does the following series converge?
QUESTION [6 upvotes]: Does the following series converge? $\sum_{n=1}^{\infty} \vert \sin n \vert ^{n}$

REPLY [6 votes]: The question has basically been answered in the comments by David Speyer and SJR.  It is a theorem of Chebyshev that that for any irrational $\alpha$ and any real $\beta$, the inequality
$$|\alpha n - k - \beta| < 3/n$$
has infinitely many solutions.  In particular, take $\alpha = 1/(2\pi)$ and $\beta = \frac12$.  Then one gets that $n$ is so close to an odd multiple of $\pi$ that $|\sin n|^n$ converges to 1 for these values.  Even if you took $|\sin n|^{n^2}$, these values would be bounded away from 0.  Certainly if the terms of a series do not converge to 0, then the series does not converge.