Does it converge?

Compute \[ \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \dots + \frac{F_k}{2^k} + \dots \] where \( F_k\) represents the the \( k^\text{th} \) term of the Fibonacci sequence: \(1, 1, 2, 3, 5, 8, 13, ... \)

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