200 Upvotes Problem

Calculus Level pending

Let \(a_{0}=\dfrac{5}{2}\) and \(a_{k}=a_{k-1}^2 -2\) for \(k \ge 1\). Compute :

\[\large \displaystyle \prod_{k=0}^{\infty} \left(1- \dfrac{1}{a_{k}} \right)\]

in closed form. Answer up to 3 decimal places

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