Progressed to infinity

Algebra Level 3

\[\begin{align} S & = \sum_{k=2}^\infty \left[ \dfrac1k + \left( \dfrac1k\right)^2 + \left( \dfrac1k\right)^3 + \cdots \right] \\ & = \left[ \dfrac12 + \left( \dfrac12\right)^2 + \left( \dfrac12\right)^3 + \cdots \right] + \left[ \dfrac13 + \left( \dfrac13\right)^2 + \left( \dfrac13\right)^3 + \cdots \right] \\ & \quad \quad + \left[ \dfrac14 + \left( \dfrac14\right)^2 + \left( \dfrac14\right)^3 + \cdots \right] + \cdots \end{align} \]

Compute the sum \(S\) above.

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