Expect Fibonacci

Calculus Level 4

Let FnF_n denote the nthn^\text{th} Fibonacci number, where F1=1,F2=1,F_1 = 1, F_2 = 1, and Fn+2=Fn+1+FnF_{n+2} = F_{n+1} + F_{n} for n=1,2,3,.n=1,2,3,\ldots.

Define fn(x)f_n (x) as a least-degree polynomial that passes through the coordinates (x,y)=(1,F1),(2,F2),(3,F3),,(n1,Fn1).(x,y)= (1, F_1), (2,F_2) , (3,F_3) , \ldots , (n-1, F_{n-1}). Hence, we define the expected Fibonacci sequence eFn^e F_n to be equal to fn(n).f_n (n).

For example, with F1=F2=1F_1 = F_2 = 1 and F3=2,F_3 = 2, we have f4(x)=12(x23x+4),f_4 (x) = \frac12\big(x^2-3x+4\big), so eF4=f4(4)=4.^e F_4 = f_4 (4) = 4.

Find the closed form of the limit below (submit your answer to three decimal places): limneFnFnFn. \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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