Solution
= Solution
Absolute convergence gives $S_n\to S$ almost surely. Put $B_j=(X_j+1)/2$. Then $S=2\sum_jB_j/2^j-1$. Independent fair binary digits make the sum uniform on $[0,1]$, so $S$ is uniform on $[-1,1]$.
= Solution
Absolute convergence gives $S_n\to S$ almost surely. Put $B_j=(X_j+1)/2$. Then $S=2\sum_jB_j/2^j-1$. Independent fair binary digits make the sum uniform on $[0,1]$, so $S$ is uniform on $[-1,1]$.