A Fibonacci series

Calculus Level 3

Let FnF_n be the nnth Fibonacci number, where F1=1F_1=1, F2=1F_2=1 and Fn+1=Fn+Fn1F_{n+1}=F_n+F_{n-1}, for n2n \geq 2. Evaluate the sum n=2FnFn+1Fn1.\sum_{n=2}^\infty \frac{F_n}{F_{n+1} F_{n-1}}.

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