Fibonacci Product

Calculus Level 5

\[\large \prod_{n=2}^\infty \left(1-\dfrac{2}{F_{n+1}^2-F_{n-1}^2+1}\right)\]

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

If the product above is equal to \( \dfrac AB\), where \(A\) and \(B\) are coprime positive integers, find \(A+B\).

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