A different type of converging sum

Calculus Level 5

\[\large \sum_{p \ \in \ P - \{1\} } p^{-1} \]

Let \(P\) be the set of perfect powers. Evaluate the sum above to 2 decimal places. If you arrive at the conclusion that the sum diverges, enter your answer as 0.

Details:

  • A positive integer \(n\) is a perfect power if it can be represented as \(m^k\) for some integers \(m > 0, k > 1\).
  • If a positive integer can be represented in multiple such ways, it still contributes to the sum only once.

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