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Homework Help: Limit of ratio of nested radicals

  1. Aug 7, 2012 #1
    1. The problem statement, all variables and given/known data
    I have a sequence an+1=sqrt(2+an) ,with a0 = sqrt(2)
    which leades to nested radicals.

    I am asked to show that for n approaching infinity:
    1) the sequence converges to 2, and that

    2) lim {(an+1 -2) / (an -2)} = 1/4

    3. The attempt at a solution
    1)I have proven the convergence to 2.
    2) for the 2nd limit, i tried de L'Hospital's rule, since we have 0 / 0 , but can't figure out exactly how to find the derivative of an (infinite) nested roots.
    - i tried also square differences , but still get to 0/0.

    Any hints?
  2. jcsd
  3. Aug 7, 2012 #2


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    Science Advisor

    Yep. Factorize [itex](a_{n+1}^2 - 4)[/itex] and compare with the same expression derived from your recurrence relation.
  4. Aug 7, 2012 #3
    Got it.
    Thanks for your help, should've noticed that =[
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