Fibonacci Summation

Calculus Level 3

$\large\sum_{n=2}^{\infty}\dfrac1{F_{n-1}F_{n+1}}$

Let $$F_n$$ denote the $$n^\text{th}$$ Fibonacci number, where $$F_0 = 0, F_1 = 1$$ and $$F_n = F_{n-1} + F_{n-2}$$ for $$n=2,3,4,\ldots$$.

Compute the sum above.

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