Let $(X_k)_{k=1}^\infty$ be a sequence of iid random variables distributed $\mathcal{N}(0,1)$. Show that $$S_n=\sum_{k=1}^n \frac{X_k}{2^k}$$ converges in $L^2$ and determine the distribution of the limit.
I tried to determine the distribution of the limit by using characteristic functions, and got that the limit $X_\infty \sim \mathcal{N}(0,\frac{1}{3})$, because $$\phi_{S_n}(t)=\phi_{\frac{X_1}{2}}(t)\cdot \phi_{\frac{X_2}{4}}(t)\cdot \dots \cdot \phi_{\frac{X_n}{2^n}}(t)=\exp \big(\frac{t^2}{2}\cdot (\underbrace{\sum_{k=1}^n \frac{1}{4^n}}_{\overset{n\to\infty}=\frac{1}{3}})\big),$$ which is the characteristic function of a RV distributed $\mathcal{N}(0,\frac{1}{3})$. But I don't know how to solve the first part...