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Homework Help: Attempt at a rigorous (dis)proof of uniform convergence

  1. Mar 3, 2008 #1
    1. The problem statement, all variables and given/known data
    1) Test the following series for Uniform Convergence on [0,1]
    [tex]
    \sum\limits_{n = 1}^{\inf } {\frac{{( - 1)^n }}{{n^{x}\ln (x)}}}
    [/tex]


    2. Relevant equations



    3. The attempt at a solution

    Obviously, it's not uniformly convergent since f(n,1) = [tex]
    \sum\limits_{n = 1}^{\inf } {\frac{{( - 1)^n }}{{n\ln (1)}}}
    [/tex] is not continuous. I'm looking for a more rigorous proof though.

    I'm wondering if thsi reasoning is correct:

    What I did was show that I can get x arbitrarily close to 1, and since log(x) is continuous, I can get arbitrarily close 1. Basically, I showed that for any 'n', I can make [tex]
    log(x) < \frac{1}{ne}
    [/tex]
    so for x sufficiently close to 1

    [tex]
    |\frac{1}{{n^{x}\ln (x)}}| > \frac{1}{|{n||\ln (x)|}} > |\frac{ne}{n}| = e
    [/tex]

    where first inequality holds because x is between 0 and 1.

    Therefore the terms of the series do not go to zero uniformly on its domain, so it can't converge.

    Is that a valid argument?
     
  2. jcsd
  3. Mar 3, 2008 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The fact that the series is not even defined at 0 and 1 is enough to conlude that it does not converge (uniformly or not) on [0,1].

    Are you sure the question asks you to discusss uniform converge on [0,1] and not (0,1) ??
     
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