Integrals I: Feynman's Trick · Superchallenge problem

Define $a_0 = 3$ and $a_{n+1} = a_n^2 - 2$. Evaluate $\displaystyle\sum_{n=0}^{\infty} \frac{1}{a_0a_1\cdots a_n}$. Express your answer in simplest radical form.