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Math Help - Power Series Representation

  1. #1
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    Unhappy Power Series Representation

    I have to find a power series representation for the following:

    1) 1/(1-x^3)
    2) 3x^2/(1-x^3)^2
    3) -3x^2/(1-x^3)
    4) ln(1-x^3)

    Well, I know that 1/(1-x) = 1 + x + x^2 + x^3 + x^4 +... = the sum of x^n
    Therefore 1/(1-x^3) = 1 + x^3 + x^6 + x^9 +...= the sum of x^(3n)

    But I don't know how to relate the others to each other they look similar, so any hins would be greatly appreciated. Thanks!
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    \frac{1}{1-x^{3}}=\sum\limits_{k=0}^{\infty }{x^{3k}}. So, why not differentiating this result and see what you get?

    As for 4), you're expected to know that \ln (1-x)=-\sum\limits_{k=1}^{\infty }{\frac{x^{k}}{k}}, then what would be a series representation for \ln(1-x^3)?
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  3. #3
    MHF Contributor
    skeeter's Avatar
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    note that #2 is the derivative of #1 ...

    #4 is the antiderivative of #3
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