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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