Fibonacci Sum

Calculus Level 5

$\sum_{n=1}^ \infty \dfrac{ n F_{n}}{ 2^n } = k{\sum_{n=1}^ \infty \dfrac{ F_{n}}{ 2^n }}$

Let $F_n$ denote the $n^\text{th}$ Fibonacci number, where $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n=2,3,4, ....$

Find the value of $k$ satisfying the equation above.

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