Powers of 2 and harmonic numbers

Calculus Level 4

Let \( H_n = 1+\dfrac12 + \dfrac13 + \cdots + \dfrac1{n}\) be the \(n^\text{th}\) harmonic number.

The sum \[ \sum_{n=1}^\infty \frac{H_n}{2^n} = \ln (a) \] for some positive integer \( a\). What is \( a\)?

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