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 \(a+b\).
by
Matt Enlow