Expect Fibonacci

Let \(F_n\) denote the \(n^\text{th} \) Fibonacci number, where \(F_1 = 1, F_2 = 1,\) and \(F_{n+2} = F_{n+1} + F_{n} \) for \(n=1,2,3,\ldots. \)

Define \(f_n (x) \) as a least-degree polynomial that passes through the coordinates \[(x,y)= (1, F_1), (2,F_2) , (3,F_3) , \ldots , (n-1, F_{n-1}).\] Hence, we define the expected Fibonacci sequence \(^e F_n \) to be equal to \(f_n (n). \)

For example, with \(F_1 = F_2 = 1\) and \(F_3 = 2,\) we have \(f_4 (x) = \frac12\big(x^2-3x+4\big), \) so \(^e F_4 = f_4 (4) = 4. \)

Find the closed form of the limit below (submit your answer to three decimal places): \[ \lim_{n\to\infty} \dfrac{ \big|^e F_n - F_n \big|}{F_n}.\]

Notation: \( | \cdot | \) denotes the absolute value function.