Let Fn denote the nth Fibonacci number, where F1=1,F2=1, and Fn+2=Fn+1+Fn for n=1,2,3,….
Define fn(x) as a least-degree polynomial that passes through the coordinates (x,y)=(1,F1),(2,F2),(3,F3),…,(n−1,Fn−1). Hence, we define the expected Fibonacci sequence eFn to be equal to fn(n).
For example, with F1=F2=1 and F3=2, we have f4(x)=21(x2−3x+4), so eF4=f4(4)=4.
Find the closed form of the limit below (submit your answer to three decimal places):
n→∞limFn∣∣eFn−Fn∣∣.
Notation: ∣⋅∣ denotes the absolute value function.