Question 1
Let $n \geqslant 1$ be an integer. For every nonempty subset $A$ of $\{1,2, \ldots, n\}$, denote by $P(A)$ the product of all the elements of $A$. For example, for $A=\{2,4,7\}$, we have $P(A)=56$. Determine the sum of the numbers $\frac{1}{P(A)}$ as $A$ ranges over all nonempty subsets of $\{1,2, \ldots, n\}$.