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Math Help - taylor series help.

  1. #1
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    Post taylor series help.

    question:

    Use integration in a term by term manner on this series for f(x)=1/(1+x) to find the taylor series for ln(1+x) about x=0

    any help would be much appreciated thanks.
    Last edited by Jonny-123; May 27th 2009 at 03:33 PM. Reason: mistakes
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  2. #2
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    Quote Originally Posted by Jonny-123 View Post
    question:

    Use integration in a term by term manner on this series for f(x)=1/(1+x) to find the taylor series for ln(1+x) about x=0

    any help would be much appreciated thanks.
    Well lets start by using there hint

    f(x)=\ln(1+x)

    Note that f(0)=ln(1)=0 we will need this later.

    Now lets take the derivative to get

    f'(x)=\frac{1}{1+x}

    This is a geometric series with r=(-1) so we get

    f'(x)=\sum_{n=0}^{\infty}(-1)^nx^n

    Now we integrate to get

    f(x)=C+\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{n+1}

    Since f(0)=0 \implies C=0 so we have


    f(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{n+1}
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