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Math Help - Alternating Sequence

  1. #1
    Member Haven's Avatar
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    Alternating Sequence

    Fix \alpha>1, take x_1 > \sqrt{\alpha} and define

    x_{n+1} = \frac{\alpha+x_n}{1+x_n} = x_n + \frac{\alpha-x_n^2}{1+x_n}

    I know this sequence should alternate about \sqrt{\alpha}. I.E., the odd terms are greater than \sqrt{\alpha} and the even terms are less than \sqrt{\alpha}. However, I cannot seem to show this.

    I've tried to use the fact that \alpha>1 and \sqrt{\alpha} < x_n but I can't seem to get anything to work.

    I would appreciate a hint very much.
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  2. #2
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    You may consider a sequence
    <br />
y_n=x_n-\sqrt{\alpha}<br />

    and

    <br />
y_n \rightarrow 0.<br />

    Now you may examine

    <br />
y_{n+1}-y_n<br />

    how it behaves around zero.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Exclamation

    Quote Originally Posted by Haven View Post
    Fix \alpha>1, take x_1 > \sqrt{\alpha} and define

    x_{n+1} = \frac{\alpha+x_n}{1+x_n} = x_n + \frac{\alpha-x_n^2}{1+x_n}

    I know this sequence should alternate about \sqrt{\alpha}. I.E., the odd terms are greater than \sqrt{\alpha} and the even terms are less than \sqrt{\alpha}. However, I cannot seem to show this.

    I've tried to use the fact that \alpha>1 and \sqrt{\alpha} < x_n but I can't seem to get anything to work.

    I would appreciate a hint very much.
    Problems of this type have been solved a lot of times, recently in...

    http://www.mathhelpforum.com/math-he...it-170087.html

    The recursive relation can be written as...

    \displaystyle \Delta_{n}= x_{n+1}- x_{n} = \frac{\alpha-x_{n}^{2}}{1+x_{n}} = f(x_{n}) (1)

    Supposing \alpha>0 there is an 'attractive fixed point' at x_{0}=\sqrt{\alpha} and the condition for monotonic convergence at x_{0} is that...

    |f(x)| \le |x_{0}-x| (2)

    It is easy to verify that the condition (2) is satisfied for 0<\alpha<1 so that in this case the sequence will be increasing if x_{1}< \sqrt{\alpha} and decreasing if x_{1}> \sqrt{\alpha}. For \alpha>1 the sequence remains convergent but is 'oscillating'...

    Kind regards

    \chi \sigma
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