Question 7

Let $$A=\{1,2,3,\ldots,17\}$$. For every nonempty subset $$B$$ of $$A$$ find the product of the reciprocals of the members of $$B$$. The sum of all such product is

Solution

For each integer $$k$$, a subset either contains $$k$$ and contributes a factor $$1/k$$ or does not contain it and contributes a factor $$1$$. Thus the sum over all subsets, including the empty subset, is $$\prod_{k=1}^{17}\left(1+\frac{1}{k}\right)=2\cdot\frac{3}{2}\cdots\frac{18}{17}=18$$. Removing the empty subset contribution $$1$$ gives $$17$$.

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Let $$A=\{1,2,3,\ldots,17\}$$. For every nonempty subset $$B$$ of $$A$$ find the product of the reciprocals of the members of $$B$$. The sum of all such product is

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