HMMT 2016 U23

Let $t=2016$ and $p=\ln 2$ . Evaluate in closed form the sum

$\sum_{k=1}^{\infty}\left(1-\sum_{n=0}^{k-1} \frac{e^{-t} t^{n}}{n!}\right)(1-p)^{k-1} p$

Dataset text NuminaMath olympiads. Quoted with source.

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1-\left(\frac{1}{2}\right)^{2016}

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