MathDB

For any nonempty set

\,S\,

of numbers, let

\,\sigma(S)\,

and

\,\pi(S)\,

denote the sum and product, respectively, of the elements of

\,S\,

. Prove that

\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),

where ``

\Sigma

'' denotes a sum involving all nonempty subsets

S

\{1,2,3, \ldots,n\}

Problem Statement