Let ${\{X_i \} }$ $i≥1$ be iid random variables with distribution $$P(X_i = 1) = P(X_i = -1) = 1/2$$
Define $Z_n = \sum_{i=1}^n \frac{X_i}{2^i}$ and let $Z = \lim_{n \to \infty} Z_n$
Find the characteristic function of $Z_n$ and $Z$
Attempt:
$\phi{_{Z_n}}(t) = \prod_{i=1}^{n} \phi{_{X_i}}(\frac{t}{2^i}) = \prod_{i=1}^{n} cos(\frac{t}{2^i})$
Simple trig calculation show that $\phi{_{Z}}(t) = sin(t)/t$
I am pretty new to characteristic functions and I am not sure if I have done it correctly. Any help is appreciated.