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Math Help - Taylor polynomial #2

  1. #1
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    Taylor polynomial #2

    Calculate General Polynomail and determine the remainder term

    g(x) = (1+x)^(1/3) , x=0

    Please help
    Last edited by mr fantastic; October 17th 2009 at 10:12 PM. Reason: Edited post title
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  2. #2
    MHF Contributor chisigma's Avatar
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    It exists a general Taylor expansion which is known as 'Binomial series' ...

    (1+x)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^{n} (1)

    ... where...

    \binom{\alpha}{n} = \frac{\alpha\cdot (\alpha-1)\cdot (\alpha-2) \dots (\alpha-n+1)}{n!} (2)

    Setting in (1) \alpha=\frac{1}{3} we obtain...

    (1+x)^{\frac{1}{3}} = 1 + \frac{x}{3} -\frac{x^{2}}{9} + \frac{5\cdot x^{3}}{81} - \dots (3)

    Kind regards

    \chi \sigma
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  3. #3
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    Thanks you!

    My professor wants (1+x)^{\frac{1}{3}} = 1 + \frac{x}{3} -\frac{x^{2}}{9} + \frac{5\cdot x^{3}}{81} - \dots<br />
put into a formula to get rid of the ... (I don't know how to do that). Also, what is the remainder?
    Last edited by mr fantastic; October 17th 2009 at 10:16 PM. Reason: Fixed the latex
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  4. #4
    MHF Contributor chisigma's Avatar
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    The Taylor expansion of a function has different 'remainders', each of them takes its name from a mathematician of ther past: for example Cauchy, Lagrange, Peano,... which is the 'remainder' that your professor requires?...

    Kind regards

    \chi \sigma
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  5. #5
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    The theorem in my book says 'Taylors Remainder theorem' (which is very confuzing)

    So I assume it is the taylor remainder
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  6. #6
    MHF Contributor chisigma's Avatar
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    Some examples of 'remainder'...

    http://mathworld.wolfram.com/LagrangeRemainder.html

    http://mathworld.wolfram.com/CauchyRemainder.html

    http://mathworld.wolfram.com/SchloemilchRemainder.html

    ... but they are not 'alone'... what is your professor's 'preference'?...

    Kind regards

    \chi \sigma
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