A Fibonacci Sum

Calculus Level 2

The value of the infinite sum

$\sum_{k=1}^\infty \frac{F_k}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{5}{32}+\frac{8}{64}+\ldots,$

where $F_k$ represents the $k^{th}$ Fibonacci number, can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive coprime integers. Find $a+b$ .