I want to know when the series $$ \sum _ { n = 1 } ^ \infty \frac { x ^ n } { \sin ( n \theta ) } $$ is convergent for $ \theta = t \pi $ such that $ t $ is irrational, e.g. $ \theta = \sqrt 3 \pi $.
It is a really interesting question because it will tell something about how the $ \sin \left( \sqrt 3 \pi n \right)$ goes to zero for some sequence goes to infinity (since $ \sqrt 3 $ is irrational sin will have a dense image in $ [ 0 , 1 ] $).
It converges when $x = 0$
$\lim_\limits{n\to\infty} \frac {x^n}{\sin n\theta}$ does not exist for $x\ne 0$