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Homework Help: Recursion relation convergence

  1. Sep 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Consider the sequence [tex]\left x_{n}\{\right\}[/tex] defined by the recursion relation,

    [tex] x_{n+1} = \frac{1}{2} \left( x_{n} + \frac{2}{x_{n}} \right) [/tex]

    where x0 > 0.

    Use the fact that "if a sequence of real numbers is monotonically decreasing and
    bounded from below, then it converges" to prove that the sequence converges.

    Show that it converges to [tex]\sqrt{2}[/tex].

    2. Relevant equations



    3. The attempt at a solution


    No idea!
    Any help would be greatly appreciated. :smile:
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 5, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    No ideas? How about the idea of doing what the hint says?

    1) Prove the sequence is increasing. Use "proof by induction" to show that [itex]x_{n+1}\ge x_n[/itex] for all n.

    2) Prove that the sequence has an upper bound. Again, use "proof by induction to show that [itex]x_n< 2[/itex] for all n.
     
  4. Sep 5, 2010 #3
    This is what I have found so far:

    [tex]x_{n+1} < x_{n}[/tex]
    [tex]\frac{1}{2}\left( x_{n} + \frac{2}{x_{n}} \right) < 2x_{n}[/tex]
    [tex]x_{n} > \sqrt{2}[/tex]

    But I have n't used induction to show that the sequence is bounded below by [tex]\sqrt{2}[/tex] :confused:

    Now, I have to show that the sequence is monotonically decreasing (by induction, as you say).

    But how I prove the [tex] x_{n+1} \leq x_{n}[/tex] for n = 1 (to begin with)?:confused:
     
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