# 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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