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Calculus Level 5

\[\large \sum _{ n=1 }^{ \infty }{ \dfrac { n{ H }_{ n } }{ { 2 }^{ n-1 } } } =A+\ln B \]

The above equation holds true for positive integers \(A\) and \(B\). Find \(A+B\).

Notation: \( H_n\) denotes the \(n^\text{th} \) harmonic number, \( H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n\).


Inspired by my own problem.

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