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Math Help - Convergence at limit points

  1. #1
    Forum Admin topsquark's Avatar
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    Convergence at limit points

    Here's another question a student hit me with tonight. (I'm not having a good night!)
    What is the radius of convergence of the sum
    \sum_{n = 0}^{\infty}\frac{(-1)^n}{ln(n)}x^n
    The answer, if I am not mistaken is |x| > 1.

    So then the student asks me, what about x = \pm 1?

    I have no answer to that. Could someone please explain how to find that out? Thanks again!

    -Dan
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by topsquark View Post
    Here's another question a student hit me with tonight. (I'm not having a good night!)

    The answer, if I am not mistaken is |x| > 1.

    So then the student asks me, what about x = \pm 1?

    I have no answer to that. Could someone please explain how to find that out? Thanks again!

    -Dan
    First, I think you want to start from n = 2, not 0 (or 1).

    x = 1 is easy. you have an alternating series which will converge (by the Cauchy alternating series test) since a_n > a_{n+1} and a_n \rightarrow 0.

    x = -1. All terms are positive. And since \frac{1}{\ln n} > \frac{1}{n} a basic comparison test will show that this series diverges.
    Last edited by mr fantastic; March 30th 2008 at 08:26 PM. Reason: Added the red stuff
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    A more interesting question is if x is a complex variable, and to find all points on the unit disk where it converges.
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