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Math Help - contractive seq

  1. #1
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    contractive seq

    let \ x_1>0 \ , \ x_{n+1}= \frac{1}{2+x_n} \ n \in N .
    prove \ that \ x_n \ is \ a \ contractive \  sequence . what \ is \ its \ limit \ ?
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  2. #2
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    \vert x_{n+1} - x_n \vert = \vert \frac{1}{2+x_n} - \frac{1}{2+x_{n-1} } \vert = \vert \frac{x_{n-1}-x_n}{(2+x_n)(2+x_{n-1}) } \vert \leq \frac{1}{4} \vert x_n-x_{n-1} \vert if x_1>0 and so it's a contractive sequence and the limit must satisfy x=\frac{1}{2+x} and you get a quadratic with only one positive root.
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  3. #3
    Senior Member sfspitfire23's Avatar
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    thx drexel
    Last edited by sfspitfire23; March 17th 2010 at 06:54 PM.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    How did you get the 1/4 from th sequence?
    x_n>0\implies \frac{1}{(2+x_n)(2+x_{n-1})}\leqslant\frac{1}{2\cdot 2}
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