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Math Help - Power Series

  1. #1
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    Power Series *edited*

    I have a Calculus Assignment on Power Series and I am very lost and confused. One of my questions looks like this:

    f(x) = 7xln(6+x)
    f(x)= CnX^n

    Find:
    C1=
    C2=
    C3=
    C4=
    and R(radius of convergence)=

    If anyone can help me out that would be great!
    Last edited by BKennedy; March 9th 2008 at 07:13 PM.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Remember that the geometric series is  \frac{1}{1-r}=\sum_{n=0}^{\infty}r^n for |r|< 1

    We need to fix up what we have so here we go...

    let g(x)=ln(x+6) so then g'(x)=\frac{1}{x+6}

    then g'(x)=\frac{\frac{1}{6}}{1-(\frac{-x}{6})}=\frac{1}{6}\sum_{n=0}^{\infty}\left(\frac{-x}{6}\right)^n=\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{6^{n+1}}

    Now integrating both sides we can get g(x)

    \int g'(x)dx= g(x) =\int \sum_{n=0}^{\infty}\frac{(-1)^nx^n}{6^{n+1}}dx=\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{6^{n+1}(n+1)}

    So

    f(x) =7x \cdot g(x)=7x \cdot \sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{6^{n+1}(n+1)}=7 \cdot \sum_{n=0}^{\infty}\frac{(-1)^nx^{n+2}}{6^{n+1}(n+1)}
    Last edited by TheEmptySet; March 9th 2008 at 09:15 PM. Reason: I solved the wrong problem lol
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