Algebraic Expression Hiding In Plain Sight

Calculus Level 2

\[ \displaystyle \sum_{n=1}^{\infty}\dfrac{2^{n}}{(2^{n}-1)(2^{n+1}-1)} = \\ \dfrac23 + \dfrac{2\times2}{3\times7} + \dfrac{2\times2\times2}{7\times15} + \cdots = \, ? \]

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