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Math Help - Two problems regarding convergence

  1. #1
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    Two problems regarding convergence

    1.

    I need to prove that (x_n)_{n\ge1} converges. Normally i'd know how to prove this (with Weierstrass theorem) but this one's a bit different :-s.

    2.

    In this problem i need to prove that (x_n)_{n\ge1} converges to 2. This translates to f_n(x_n)=x^{n}_{n}+\ln{x_n}=2^n  \to  x^{n}_{n}+\ln{x_n}-2^n=0
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    For the first one...

    I think you should "do" Newton-Raphson to f(x)
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  3. #3
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    I don't know what that is... never seen it in high-school :-s (or either i don't know it by this name)
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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  5. #5
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    i did x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} and end up with x_{n+1}=\frac{x^{2}_{n}}{\arctan{x_n}+x^{2}_{n}\ar  ctan{x_n}} i don't really know what to do with it

    And i can't use this because:

    1. i can't use theorems or methods they don't teach at high-school for my exam
    2. it's hard for me to understand without the guidance of a teacher.
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  6. #6
    MHF Contributor chisigma's Avatar
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    The first difference equation can be written in the equivalent form...

    \Delta_{n} = x_{n+1} - x_{n} = x_{n}\ (\tan^{-1} x_{n} -1)= \varphi(x_{n}) (1)

    The function \varphi(*) is represented here...



    It has an attractive fixed point in x=0 and a repulsive fixed point at x=- \frac{\pi}{4} so that any value - 1 < x_{0} < \frac{\pi}{4} will produce a sequence converging at x=0...

    Kind regards

    \chi \sigma
    Attached Thumbnails Attached Thumbnails Two problems regarding convergence-mhf65.bmp   Two problems regarding convergence-mhf65.jpg  
    Last edited by chisigma; June 27th 2010 at 06:09 AM. Reason: restored image
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