USAMO 1991 Problem 2

For each non-empty subset of $\{1,2, \ldots, n\}$ take the sum of the elements divided by the product. Show that the sum of the resulting quantities is $n^{2}+2 n-(n+1) s_{n}$, where $s_{n}=1+1 / 2+1 / 3+\ldots+1 / n$.