What is the limit of the recurrence relation for g_n?

[Marian123]

Hi all

Suppose that , a_{n+1}=a_n^2-2 and g_n=\frac{a_1a_2...a_n}{a_{n+1}}.
Evaluate \lim_{n\rightarrow \infty } g_n.

I have seen some information in this link. Besides, the sequence gn seems as a good rational approximation for \sqrt5. I know that the answer is 1, But I can't find any nice solution. Any hint is strongly appreciated.

 

[jasonRF]

I'm not sure the limit exists in general ...

If $$a_1=0$$ then $$g_n=0$$ for all n, so the limit is zero. If $$a_1=1$$, then g is an alternating sequence and the limit does not exist. If $$a_1=2$$, then $$g_n=2^{n-1}$$ which diverges. In fact, after playing for a few minutes, I can only get g to either converge to zero or not converge at all.

jason