MathDB

Consider all non-empty subsets of the set

\{1,2\cdots,n\}

. For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as

S_n

. For example,

S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3}

(i) Show that

S_n=\frac1n+\left(1+\frac1n\right)S_{n-1}

.
(ii) Hence or otherwise, deduce that

S_n=n

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