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 \(n=2,3,4, ....\)