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Math Help - Two Quick Questions - Power Series

  1. #1
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    Two Quick Questions - Power Series

    Hello all; having some trouble with these last homework questions!

    Instructions: Find a power series representation for the function and determine the radius of convergence.

    #1) f[x] = x^2/(1-2x)^2

    Should I use partial fractions to split this up?

    #2) Evaluate the indefinite integral as a power series. What is the radius of convergence. Integral of ArcTan(x^2)dx

    Thanks so much for your help in the past and future all!
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  2. #2
    MHF Contributor chisigma's Avatar
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    Because ...

    \frac{1}{1-2x}= \sum_{n=0}^{\infty} (2x)^{n} , |x|<\frac{1}{2} (1)

    ... and...

    \frac{d}{dx} \frac{1}{1-2x} = \frac{2}{(1-2x)^{2}} (2)

    ... is...

    \frac{x^{2}}{(1-2x)^{2}}= \sum_{n=0}^{\infty} \frac{2^{n-1}}{n+1}\cdot x^{n+3} , |x|<\frac{1}{2} (3)

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by Sprintz View Post
    Hello all; having some trouble with these last homework questions!

    Instructions: Find a power series representation for the function and determine the radius of convergence.

    #1) f[x] = x^2/(1-2x)^2

    Should I use partial fractions to split this up?

    As \frac{1}{1-2x}=1+2x+4x^2+...+(2x)^n+...\,\,whenever\,\,|x|<\f  rac{1}{2}\,,\,\,and\,\,since\,\, \frac{2}{(1-2x)^2}=\left(\frac{1}{1-2x}\right)'\,,\,\,then...

    #2) Evaluate the indefinite integral as a power series. What is the radius of convergence. Integral of ArcTan(x^2)dx

    \arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-...\,,\,\,so \,\,\arctan x^2=x^2-\frac{x^6}{3}+...

    Tonio

    Thanks so much for your help in the past and future all!
    .
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