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Math Help - General Binomial Theorm

  1. #1
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    General Binomial Theorm

    I just need to verify my answer to the following problem

    Use the General Bionomial Theorem to determine first four terms of the Taylor series at 0 for the function

    f(x) = (1-2x)^-1/3

    I have

    By the General Bionomial Theorem

    (1-2x)^-1/3

    (-1/3)
    ( n ) x^n, for |x|<1

    Where
    (-1/3)
    ( n ) = (-1/3)(-4/3)(-7/3)....(-1/3-n=1) / n!

    hence
    (1-2x)^-1/3

    = 1 + (-1/3)/1 -2x + (-1/3)(-4/3)/2! -2x^2 + (-1/3)(-4/3)(-7/3)/3! -2x^3 .....

    = 1+ 2/3x - 4/9x^2 + 28/81x^3 ........

    I also need to state the radius of convergence of this power series.

    can anyone help?
    Last edited by Arron; July 27th 2011 at 12:39 AM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: General Binomial Theorm

    In general, the radius of convergence for the series \sum_{n=0}^{+\infty}\binom{p}{n}t^n\;(p\in\mathbb{  R}) is 1 (why?). Now, substitute t=-2x .
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: General Binomial Theorm

    Quote Originally Posted by Arron View Post
    I just need to verify my answer to the following problem

    Use the General Bionomial Theorem to determine first four terms of the Taylor series at 0 for the function

    f(x) = (1-2x)^-1/3

    I have

    By the General Bionomial Theorem

    (1-2x)^-1/3

    (-1/3)
    ( n ) x^n, for |x|<1

    Where
    (-1/3)
    ( n ) = (-1/3)(-4/3)(-7/3)....(-1/3-n=1) / n!

    hence
    (1-2x)^-1/3

    = 1 + (-1/3)/1 -2x + (-1/3)(-4/3)/2! -2x^2 + (-1/3)(-4/3)(-7/3)/3! -2x^3 .....

    = 1+ 2/3x - 4/9x^2 + 28/81x^3 ........

    I also need to state the radius of convergence of this power series.

    can anyone help?
    taylor &#40;1-2x&#41;&#94;-1&#47;3 at x&#61;0 - Wolfram|Alpha
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  4. #4
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    Re: General Binomial Theorm

    What does that mean?
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  5. #5
    Flow Master
    mr fantastic's Avatar
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    Re: General Binomial Theorm

    Quote Originally Posted by Arron View Post
    What does that mean?
    What does what mean? Please quote what you refer to in a thread.

    If you're referring to post #3, then what it means is that you were shown what the answer is so that you can see whether your answer is correct (and if you're answer is not correct, then you can see what term is wrong and go back to your working and look for the mistake(s)).

    If you're referring to post #2, then you need to be much more specific with what you don't understand.
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