Show that if n is a positive integer, then (Here the sum

Chapter 5, Problem 65E

(choose chapter or problem)

Show that if $n$ is a positive integer, then

$\sum_{\left\{a_{1}, \ldots, a_{k}\right\} \subseteq\{1,2, \ldots, n\}} \frac{1}{a_{1} a_{2} \cdots a_{k}}=n$ .

(Here the sum is over all nonempty subsets of the set of the $n$ smallest positive integers.)

Equation Transcription:

$n$

$\sum\limits_{\{{a}_{1},\ .....,{a}_{k}\}\ \subseteq{}\ \{\ 1,2,\ .....n\}}^{\ }\ \frac{1}{{a}_{1}{a}_{2}\cdot{}\cdot{}\cdot{}{a}_{k}}\ =\ n$

$n$

Text Transcription:

\sum_{\{a_{1}, \ldots, a_{k}\} \subseteq\{1,2, \ldots, n\}} \frac{1}{a_{1} a_{2} \cdots a_{k}}=n .

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