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Math Help - Ratio test to find the interval of convergence of a MacLaurin Series

  1. #1
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    Smile Ratio test to find the interval of convergence of a MacLaurin Series

    Hello,

    How would you find the interval of convergence for the MacLaurin series of f(x) = loge(1-x), which I worked out to be: x - 1/2x^2 + 1/3x^3 - 1/4x^4 + 1/5x^5.

    I know we can use the ratio test to find the interval of convergence, but I don't know how that's supposed to be done, are we meant to be done with a MacLaurin series.

    Thankyou, I really appreciate your help.
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    TD!
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    Re: Ratio test to find the interval of convergence of a MacLaurin Series

    Are you sure you don't mean a series for ln(1+x) (instead of the minus sign)? In any case, say you found the power series:

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

    There are formules to calculate the radius of convergence, see here for example. In your case, compute the limit:

    \lim_{n\to\infty} \left| \frac{\displaystyle \frac{(-1)^{n+1}}{n}}{\displaystyle \frac{(-1)^{n+2}}{n+1}} \right| = \ldots

    You'll find the radius but that doesn't give you the interval just yet, you'll have to check the end points (the boundary) separately.
    Thanks from happyfingerrs
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