Binet's Formula is a closed form expression for Fibonacci numbers:
It can be said that when , however, when .
If we plot in the complex plane for , we get an oscillating curve which intersects the real axis at the Fibonacci numbers, as shown in the picture above.
Let denote the area under a segment between two consecutive Fibonacci numbers ( and ).
Find .
Note: denotes the golden ratio .
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Binet's Formula can be rewritten as
5 φ n − ( − φ ) − n = 5 φ n − 5 φ − n cos n π + 5 φ − n i sin n π
Since the curve between two integers is either completely negative or positive, we can define A n as
A n = ∣ ∣ ∣ ∣ ∫ n n + 1 ( 5 φ − t sin t π ) d ( 5 φ t − 5 φ − t cos t π ) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ n n + 1 ( 5 φ − t sin t π ) ( 5 φ t ln φ + 5 φ − t ln φ cos t π + 5 π φ − t sin t π ) d t ∣ ∣ ∣ ∣ = 5 1 ∣ ∣ ∣ ∣ ∫ n n + 1 ( ln φ sin t π + φ − 2 t ln φ sin t π cos t π + φ − 2 t π sin 2 ( t π ) ) d t ∣ ∣ ∣ ∣ Consider 0 ≤ ∣ sin t π cos t π ∣ ≤ 1 And ∣ ∣ ∣ ∣ ∣ ∫ a b f ( t ) d t ∣ ∣ ∣ ∣ ∣ ≤ ∫ a b ∣ f ( t ) ∣ d t Applying both inequalities, we have φ − 2 t ∣ sin t π cos t π ∣ ≤ φ − 2 t ∣ ∣ ∣ ∣ ∫ n n + 1 φ − 2 t sin t π cos t π d t ∣ ∣ ∣ ∣ ≤ ∫ n n + 1 ∣ ∣ φ − 2 t sin t π cos t π ∣ ∣ d t ≤ ∫ n n + 1 φ − 2 t d t ∣ ∣ ∣ ∣ n → ∞ lim ∫ n n + 1 φ − 2 t sin t π cos t π d t ∣ ∣ ∣ ∣ ≤ n → ∞ lim ∫ n n + 1 φ − 2 t d t = n → ∞ lim − 2 1 ( φ − 2 ( n + 1 ) − φ − 2 n ) = 0 By Squeeze Theorem, we get ∣ ∣ ∣ ∣ n → ∞ lim ∫ n n + 1 φ − 2 t sin t π cos t π d t ∣ ∣ ∣ ∣ = 0 n → ∞ lim ∫ n n + 1 φ − 2 t sin t π cos t π d t = 0 Using the same argument, we can show that n → ∞ lim ∫ n n + 1 φ − 2 t sin 2 ( t π ) d t = 0
Hence n → ∞ lim A n = n → ∞ lim 5 1 ∣ ∣ ∣ ∣ ∫ n n + 1 ( ln φ sin t π + φ − 2 t ln φ sin t π cos t π + φ − 2 t π sin 2 ( t π ) ) d t ∣ ∣ ∣ ∣ = 5 ln φ n → ∞ lim ∣ ∣ ∣ ∣ ∫ n n + 1 sin t π d t ∣ ∣ ∣ ∣ = 5 π − ln φ n → ∞ lim ∣ ( cos ( n + 1 ) π − cos n π ) ∣ = 5 π 2 ln φ n → ∞ lim ∣ cos n π ∣ For n ∈ Z 0 + , this gives n → ∞ lim A n = 5 π 2 ln φ