Question 4
Let $n$ be a positive integer. Maurice writes on one line all the $2^{n}-1$ non-empty subsets of the set $\{1,2, \ldots, n\}$. Then, below each subset, he writes the product of its elements. Finally, he writes the reciprocals of the numbers in the second line and computes their sum. What will be the value of the sum (as a function of $n$) that Maurice obtains?
Example: for $n=3$, Maurice obtains
| $\{1\}$ | $\{2\}$ | $\{3\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 2 | 3 | $1 \cdot 2=2$ | $1 \cdot 3=3$ | $2 \cdot 3=6$ | $1 \cdot 2 \cdot 3=6$ |
| $\frac{1}{1}+$ | $\frac{1}{2}+$ | $\frac{1}{3}+$ | $\frac{1}{2}+$ | $\frac{1}{3}+$ | $\frac{1}{6}+$ | $\frac{1}{6}=3$. |
Example: for $n=3$, Maurice obtains
| $\{1\}$ | $\{2\}$ | $\{3\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 2 | 3 | $1 \cdot 2=2$ | $1 \cdot 3=3$ | $2 \cdot 3=6$ | $1 \cdot 2 \cdot 3=6$ |
| $\frac{1}{1}+$ | $\frac{1}{2}+$ | $\frac{1}{3}+$ | $\frac{1}{2}+$ | $\frac{1}{3}+$ | $\frac{1}{6}+$ | $\frac{1}{6}=3$. |