Does the following series converge? $\sum_{n=1}^{\infty} \vert \sin n \vert ^{n}$
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. 





$\alpha xy\beta<3/x$
has infinitely many integer solutions. From this follows easily that $n$ is infinitely often as close to an odd multiple of $\pi/2$ as David's argument requires. – SJR Feb 8 '11 at 14:41