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Math Help - Monotone sequences and Cauchy sequences

  1. #1
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    Monotone sequences and Cauchy sequences

    Suppose that x>0. Define a sequence (s_n) by s_1=k, and s_(n+1) = (((s_n)^2)+x)/(2s_n) for n elements in N. Prove that for any k>0, lim s_n = sqrt(x).

    I've gotten that I need to show that (s_(n+1))^2)-x = (((s_n)^2)-x)^2/(4(s_n)^2)=>0, so that s_n=>sqrt(x) for n=>2.

    Also I need to prove that (s_n) is decreasing for n=>2, by showing that (s_n)-(s_(n+1))>=0.

    Unfortunately I don't know how to exactly show these two steps.

    Any help would be great.
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  2. #2
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    We will assume n\geq 2 and so s_n = \tfrac{1}{2}(s_{n-1} + xs_{n-1}^{-1}). Therefore, s_n^2 = \tfrac{1}{4}(s_{n-1}^2 + 2x + x^2s_{n-1}^{-2}). We get from here s_n^2 = x + \tfrac{1}{4}(s_{n-1} - xs_{n-1}^{-1})^2 \geq x \implies s_n \geq \sqrt{x}. This was your first half of your problem.

    Now, s_{n+1} - s_n = \tfrac{1}{2} (s_n + xs_{n}^{-1}) - s_n = \tfrac{1}{2}(xs_n^{-1} - s_n) < \tfrac{1}{2}(s_n - s_n) = 0.
    (The last inequality follows because s_n^2 > x \implies s_n > xs_n^{-1})
    Last edited by ThePerfectHacker; March 21st 2009 at 08:58 PM.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    We will assume n\geq 2 and so s_n = \tfrac{1}{2}(s_{n-1} + xs_{n-1}^{-1}). Therefore, s_n^2 = \tfrac{1}{4}(s_{n-1}^2 + 2x + x^2s_{n-1}^{-2}). We get from here s_n^2 = x + (s_{n-1} - xs_{n-1})^2 \geq x \implies s_n \geq \sqrt{x}. This was your first half of your problem.

    Now, s_{n+1} - s_n = \tfrac{1}{2} (s_n + xs_{n}^{-1}) - s_n = \tfrac{1}{2}(xs_n^{-1} - s_n) < \tfrac{1}{2}(s_n - s_n) = 0.
    (The last inequality follows because s_n^2 > x \implies s_n > xs_n^{-1})
    We have :


    s_n^2 = \tfrac{1}{4}(s_{n-1}^2 + 2x + x^2s_{n-1}^{-2}).

    AND

    s_n^2 = x + (s_{n-1} - xs_{n-1})^2.

    But . \tfrac{1}{4}(s_{n-1}^2 + 2x + x^2s_{n-1}^{-2}),is not equal to
    x + (s_{n-1} - xs_{n-1})^2 as your calculations show .

    Please explain your calculations ,i am completely confused
    Last edited by xalk; March 21st 2009 at 05:09 PM. Reason: wrong post
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  4. #4
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    Quote Originally Posted by xalk View Post
    We have :


    s_n^2 = \tfrac{1}{4}(s_{n-1}^2 + 2x + x^2s_{n-1}^{-2}).

    AND

    s_n^2 = x + (s_{n-1} - xs_{n-1})^2.

    But . \tfrac{1}{4}(s_{n-1}^2 + 2x + x^2s_{n-1}^{-2}),is not equal to
    x + (s_{n-1} - xs_{n-1})^2 as your calculations show .

    Please explain your calculations ,i am completely confused
    I fixed my post now, there was a mistake in it.
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