Problem 29 (difficulty: 7/10)

Consider the set \(\displaystyle H:=\{2,3,\ldots n+1\}\). Prove that

\(\displaystyle \sum_{\emptyset\neq S\subset H}\prod_{i\in S}\frac1i=n/2. \)

(For example for \(\displaystyle n=3\) we have \(\displaystyle {1\over 2}+\frac13+{1\over 4}+{1\over 2\cdot 3}+{1\over 2\cdot 4}+ {1\over 3\cdot 4}+{1\over 2\cdot 3\cdot 4}={3\over 2}\)).

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