Algebraic 1

Algebra Level 4

Let \(k\) be a natural number. Define \(S_k\) as the sum of the infinite geometric series with first term \(k^2 - 1\) and ratio \(\dfrac{1}{k}\), that is \(S_k = \dfrac{k^2 - 1}{k^0} + \dfrac{k^2 - 1}{k^1} + \dfrac{k^2 - 1}{k^2} + \cdots\). Find the value of

\[\displaystyle \sum_{k=1}^\infty \frac{S_k}{2^{k-1}} .\]

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