<s>[INST] Sum of the recursive series Let $\langle a_n \rangle$ be a recursive sequence given by $a_1>2$ and,
$a_{n+1}=a_n^2-2$  for $n \in \mathbb N$
Show that
$\sum_{n=1}^\infty \frac{1}{a_1a_2\cdots a_n} = \frac{a_1-\sqrt{a_1^2-4}}{2}$
I have reached this step:
$\frac{1}{a_1a_2a_3\cdots a_n}=\frac{1}{2} (\frac{a_n}{a_1a_2\cdots a_n-1} -\frac{a_n-1}{a_1a_2\cdots a_n})$
But I am not able to obtain the final expression .Please help me to obtain it. Thanks for help in advance.
 [/INST] Notice the function $x \mapsto  x^2 - 2$ send $(2,\infty)$ to $(2,\infty)$.
Start from $a_1 > 2$, it is easy to see $a_n > 2$ for all $n$. 
For each $n$, pick a $x_n > 1$ such that $a_n = x_n + \frac{1}{x_n}$, the recurrence relation becomes
$$x_{n+1} + \frac{1}{x_{n+1}} = \left(x_n + \frac{1}{x_n}\right)^2 - 2
= x_{n}^2 + \frac{1}{x_{n}^2}\quad\implies\quad x_{n+1} = x_n^2$$
Let $\lambda = x_1$. Solving above recurrence relation gives us
$$x_n = \lambda^{2^{n-1}}\quad \implies\quad a_n = \lambda^{2^{n-1}} + \lambda^{-2^{n-1}}
= \frac{\lambda^{2^n} - \lambda^{-2^n}}{\lambda^{2^{n-1}} - \lambda^{-2^{n-1}}}
$$
The summands in the sum are telescoping products.
$$\begin{align}\prod_{k=1}^n \frac{1}{a_k} &= \prod_{k=1}^n 
\frac{\lambda^{2^{k-1}} - \lambda^{-2^{k-1}}}{\lambda^{2^{k}} - \lambda^{-2^k}}
= \frac{\lambda - \lambda^{-1}}{\lambda^{2^n} - \lambda^{-2^n}}\\
&=  \frac{\lambda^2-1}{\lambda}\frac{\lambda^{2^n}}{(\lambda^{2^n})^2 - 1}
= \frac{\lambda^2-1}{\lambda}\left[\frac{1}{\lambda^{2^n}-1} - \frac{1}{\lambda^{2^{n+1}} - 1}\right]
\end{align}
$$
The sum itself is also a telescoping one. At the end, we have
$$\begin{align}\sum_{n=1}^\infty \prod_{k=1}^n \frac{1}{a_k}
&= \frac{\lambda^2-1}{\lambda} \sum_{n=1}^\infty
\left[\frac{1}{\lambda^{2^n}-1} - \frac{1}{\lambda^{2^{n+1}} - 1}\right]
= \frac{1}{\lambda}\\
&= \frac12\left(
(\lambda + \lambda^{-1}) - (\lambda - \lambda^{-1})
\right) 
=
\frac12\left(
(\lambda + \lambda^{-1}) - \sqrt{(\lambda + \lambda^{-1})^2 - 4}
\right)\\ 
&= \frac{a_1 - \sqrt{a_1^2-4}}{2}
\end{align}
$$
</s>