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The recurrence relation

  1. Oct 19, 2011 #1
    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.
     
  2. jcsd
  3. Oct 19, 2011 #2

    jasonRF

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    Gold Member

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

    If [tex]a_1=0[/tex] then [tex]g_n=0[/tex] for all n, so the limit is zero. If [tex]a_1=1[/tex], then g is an alternating sequence and the limit does not exist. If [tex]a_1=2[/tex], then [tex]g_n=2^{n-1}[/tex] 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
     
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