Powers of 2 and harmonic numbers

Calculus Level 3

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 \( a\)?