Let $z=\sqrt(3)\frac{5}{12}+i\frac{5}{12}$ and let $b\in \mathbb{R}$. Define $x_n=(b\cdot z)^n$. Now, I know that
if $|bz|<1$, then $x_n \to 0$
if $|bz|>1$, then $|x_n| \to \infty.$
if $|bz|=1$, then $|x_n|=1$ for all $n$, hence $(x_n)$ is bounded.
and that $|z|=5/6$
Question
What does the second and third case actually tell me about $x_n$. I only know that if $|x_n|\to 0$ when $n\to \infty$ then so does $x_n$.
For case 2; If $|x_n|\to \infty$ as $n \to \infty$ then what happens to $x_n$? Would that also diverge or do we actually not know this for sure. Is there some property for my sequence that allows me to draw some conclusion from this?
For case 3; If $|x_n|=1$ for all $n$ then $x_n$ should also be $1$ since we are taking powers of $1$ i.e $1^n$. That kind of makes sense for case 3.