# Expect Fibonacci

Calculus Level 4

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.

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