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Homework Help: Recursive sequence convergence

  1. Oct 15, 2007 #1
    1. The problem statement, all variables and given/known data

    Let [tex]x_1 < x_2[/tex] be arbitrary real numbers and let [tex]x_n :=\frac{1}{3}x_{n-1} + \frac{2}{3}x_{n-2}[/tex]. Prove the sequence [tex](x_n)[/tex] converges.

    2. Relevant equations

    Since this problem comes from the section on Cauchy sequences, I assume we will need to show [tex](x_n)[/tex] is a Cauchy sequence. I'm not so well-versed in working with the recursive sequences especially with arbitrary initial values.

    Any advice on getting started?
     
  2. jcsd
  3. Oct 15, 2007 #2
    would this be a valid solution? it looks like i can show the sequence is contractive.

    [tex]|x_{n+1}-x_n| = |\frac{1}{3}x_n + 2 x_{n-1} - x_n | = \frac{2}{3}|x_{n-1} - x_n|[/tex]

    Thus [tex](x_n)[/tex] is contractive, so it is convergent.
     
    Last edited: Oct 15, 2007
  4. Oct 15, 2007 #3
    OK. so now how do we go about finding the limit of [tex](x_n)[/tex]?
     
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