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Math Help - Power series representation.

  1. #1
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    Power series representation.

    Hi.

    Find the power series representation of:

    f(x) = x/9-x^2

    factor out x/9 and we get:

    1/(1-x^2/9) or 1/(1-(x/3)^2)

    Now my question is. The solution shows that we use the geometric series representation, 1/1-r. In this case, shouldn't r be x^2/9. The solution uses x/3.
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  2. #2
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    Quote Originally Posted by Kuma View Post
    Now my question is. The solution shows that we use the geometric series representation, 1/1-r. In this case, shouldn't r be x^2/9. The solution uses x/3.
    It doesn't matter! You could let r = \frac{x}{3} or r = x^2/9 and both would give you the answer:

    \displaystyle \frac{1}{1-\frac{x^2}{9}} = \sum_{k = 0}^{\infty}\left(\frac{x^2}{9}\right)^k = \sum_{k=0}^{\infty}\left(\frac{x^2}{3^2}\right)^k = \sum_{k=0}^{\infty}\left(\frac{x}{3}\right)^{2k} and \displaystyle \frac{1}{1-\left(\frac{x}{3}\right)^2} = \sum_{k=0}^{\infty}\left(\frac{x}{3}\right)^{2k}

    This is because in general \displaystyle \frac{1}{1-x^n} = \sum_{k=0}^{\infty}x^{nk} (you can see why by simply letting x^n = r).
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