HMMT 2005 U3

Find the sum

$\frac{2^{1}}{4^{1}-1}+\frac{2^{2}}{4^{2}-1}+\frac{2^{4}}{4^{4}-1}+\frac{2^{8}}{4^{8}-1}+\cdots .$

Dataset text NuminaMath olympiads. Quoted with source.

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