Convergence of a Sequence for a real number $b$

Question

Let $z=\sqrt(3)\frac{5}{12}+i\frac{5}{12}$ and let $b\in \mathbb{R}$. Define $x_n=(b\cdot z)^n$.

• Determine all real $b$ where $\left \{ x_n \right \}_{n \in \mathbb{N}}$ is convergent
• Determine all real $b$ where $\left \{ x_n \right \}_{n \in \mathbb{N}}$ has a convergent subsequence

How do I even start to argue for this? I know that any bounded complex sequence has a convergent subsequence but that is not useful as I need to handle problem one first. I have found the argument of $z$ and the modulus of $z$.

Answer 1

Hints:

1. if $|bz|<1$, then $x_n \to 0$

2. if $|bz|>1$, then $|x_n| \to \infty.$

3. if $|bz|=1$, then $|x_n|=1$ for all $n$, hence $(x_n)$ is bounded.