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Using the Weierstrass M-test to check for uniform convergence

  1. Mar 28, 2010 #1
    1. The problem statement, all variables and given/known data

    [tex]\text{Show that the series }f(x) = \sum_{n=0}^\infty{\frac{nx}{1+n^4 x^2}}\text{ converges uniformly on }[1,\infty].[/tex]
    2. Relevant equations



    3. The attempt at a solution

    I think we should use the Weierstrass M-test, but I'm not sure if I'm applying it correctly:

    For [tex]x \in [1,\infty)[/tex] we have

    [tex]\frac{nx}{1+n^4 x^2} \leq \frac{n}{n^4} = \frac{1}{n^3}[/tex]

    and [tex]\sum_{n=1}^\infty {\frac{1}{n^3}}[/tex] is a convergent p-series.

    Also, for n=0, we have the series function is equal to 0, so this will not affect the convergence of the series.

    Therefore, by the Weierstrass M-test, the original series is uniformly convergent.

    Have I got it right? Thanks in advance.
     
  2. jcsd
  3. Mar 30, 2010 #2
    Your estimate is right : nx/(1 + n^4* x^2) attains a maxima equal to 1/2n
    (at x = 1/n^2) & decreases there onwards.
     
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