  Consider two series $$\sum_{n=1}^\infty\frac{\sin^n\pi\theta
n}{n^\alpha},\quad\sum_{n=1}^\infty\frac{\cos^n\pi\theta n}{n^\alpha}.$$ We
show that number-theoretical properties of $\theta$ have a strong effect on the
convergence when $0<\alpha\leq 1$. The complete investigation for
$\theta\in\mathbb Q$ is given. For irrational $\theta$ we prove the result
which depends on how well $\theta$ can be approximated with rational numbers,
i.e. on its irrationality measure. We obtain that if $\alpha>\frac12$ then both
series converge absolutely for almost all real $\theta$. Finally, we construct
such an everywhere dense set of $\theta$ that both series diverge when
$\alpha\leq 1$.
