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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