Find the limit \begin{equation*} \lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right] \end{equation*} where $\lambda > 0$.
My guess is that the limit is equal to $\lambda$.
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