$\begin{eqnarray*}S_n^1 &=&{1\over 2}+{2\over 2^2}+\ldots+{n\over 2^n}\\
1-{1\over 2^n} &=&{1\over 2}+{1\over 2^2}+\ldots+{1\over 2^n}\end{eqnarray*}$