A Fibonacci Sum

Calculus Level 2

The value of the infinite sum

k=1Fk2k=12+14+28+316+532+864+,\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 FkF_k represents the kthk^{th} Fibonacci number, can be written as ab\frac{a}{b}, where aa and bb are positive coprime integers. Find a+ba+b.

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