2014 Putnam A3

Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute

$$

\prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right)

$$

in closed form.

设 $a_0 = 5/2$ 和 $a_k = a_{k-1}^2 - 2$ 对于 $k \geq 1$。计算

$$

\prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right)

$$

以封闭形式。