Powers of 2 and harmonic numbers

Calculus Level 3

Let \( H_n = 1+\frac12 + \frac13 + \cdots + \frac1{n}\) be the \(n^\text{th}\) harmonic number.

The sum \(\displaystyle \sum_{n=1}^\infty \frac{H_n}{2^n} = \ln (a) \) for some positive integer \( a.\)

What is \( a?\)

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