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Proving convergence of recursive sequence.

  1. Mar 23, 2014 #1
    1. The problem statement, all variables and given/known data

    Prove for c>0 the sequence [itex]{x_n} = \frac{1}{2}(x_{n-1} + \frac{c}{x_{n-1}})[/itex] converges.

    3. The attempt at a solution

    This is proving difficult, I have never dealt with recursive sequences before. Any help would be appreciated. Thanks.
  2. jcsd
  3. Mar 23, 2014 #2
    So, I have an idea, but i'm not sure if it proves convergence.

    If [itex]x_{n} = g(x_{n-1})[/itex] then all I have to do it let [itex]x = g(x)[/itex].

    EDIT: Upon further thought, it does not as convergence => [itex]x_{n} = g(x_{n-1})[/itex]. It is not a iff situation. Back at square one.
    Last edited: Mar 23, 2014
  4. Mar 23, 2014 #3
    You have the following handy theorem:

    I would start by trying to apply this one. Can you show the sequence is increasing/decreasing? Can you find a lower/upper bound?
  5. Mar 23, 2014 #4
    This is a nice remark, because if the sequence converges, you can find out the limit point this way. Indeed, you have ##x_n = g(x_{n-1})##. Thus

    [tex]x = \lim_{n\rightarrow +\infty} x_n = \lim_{n\rightarrow +\infty} g(x_{n-1}) = g(x)[/tex]

    So can you figure out what the limit point ##x## is? (of course you still have to prove it actually converges!)
  6. Mar 23, 2014 #5
    So this is what I have so far.

    [itex] x_0 = r [/itex] r is a non-zero real number

    [itex] x_{n+1} = x_n(\frac{1}{2} + \frac{c}{2x_n^2}) [/itex]

    Case 1: ##r^2 < c~ \Rightarrow## sequence is increasing

    Case 2: ##r^2 > c ~\Rightarrow## sequence is decreasing
    Last edited: Mar 23, 2014
  7. Mar 23, 2014 #6


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    Not quite:

    Suppose c = 4 and x_0 = 1, then x_1 = 5/2 and x_2 = 89/40, so neither increasing nor decreasing!

    You're on the right track, though.
  8. Mar 23, 2014 #7

    Ray Vickson

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    To see what is happening, look at the graph of
    [tex] y = f(x), \text{ where } f(x) = \frac{1}{2} \left( x + \frac{c}{x} \right)[/tex]
    Look up material about "cobweb plots"; see, eg., http://en.wikipedia.org/wiki/Cobweb_plot . This will give you some intuition about what is happening; it is not yet a 'proof', but may help get you started.
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