For an interesting example, take $\sum_{n=1}^\infty \frac{|\sin(n)|^n}{n}$.
Deciding whether or not this converges seems to require more knowledge than is currently available about the rational approximations of $\pi$.  The series
$\sum_{n=1}^\infty \frac{|\sin(n t \pi)|^n|}{n}$ converges for almost every 
real $t$ (in the sense of Lebesgue measure), but diverges for $t$ in a dense $G_\delta$ subset of $\mathbb R$.