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 $$a+b$$.

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