Jasper's sum of subset products

Algebra Level 4

For each non-empty subset of integers \( \{1,2\cdots,n\} \), consider the reciprocal of the product of the elements. Let \(S_n\) denote the sum of these products. For example, \[ S_3=\frac{1}1+\frac{1}2+\frac{1}3+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 3}+\frac{1}{2\cdot 3}+\frac{1}{1\cdot 2\cdot 3}. \]

Find the sum of digits of \( S_{2013} \).

This problem is shared by Jasper S.

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