My favourite summation

Calculus Level 4

\[\large \sum_{x\in S}\frac{1}{x-1}=\frac{1}{3}+\frac{1}{7}+\frac{1}{8}+\frac{1}{15}+\frac{1}{24}+\ldots\]

Find the value of the above summation when it's summed over all elements in set \(S\).

Set \(S\) consists of all perfect powers excluding 1 and excluding duplicates.

\[S=\{4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100,...\}\]

• A perfect power is a number of the form \(m^n\) where \(m\) and \(n\) are natural numbers and \(n\neq 1\).

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