Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in distribution and determine the limiting distribution. Also need to determine if $X_n$ converges in other senses.
I know in general, the Bernoulli Characteristic function is \begin{align} \varphi(t) = 1 - p + pe^{it} \end{align}
But beyond that, I am really lost on how to use it to show convergence.