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Convergence of serie

  1. Apr 4, 2005 #1

    quasar987

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    I need help with the following problem.

    Consider the serie of function

    [tex]\sum_{n=1}^{\infty}\frac{1}{1+n^2x}[/tex]

    The serie is undefined for [itex]x \in \{0\}\cup \{-1/n^2, \ n\in \mathbb{N}\}[/itex]. I want to find wheter it converges pointwise in (-1, 0) or not and if it does, does it converge uniformly?

    The way I would start this problem is by saying: For a given number [itex]m \in \mathbb{N}[/itex], consider

    [tex]x_0 \in \left(\frac{-1}{m^2} \ ,\frac{-1}{(m+1)^2}\right)[/tex]

    Consider

    [tex]f_n(x) = \frac{1}{1+n^2x}[/tex]

    Then

    [tex]|f_n(x_0)| = \frac{1}{|1+n^2x_0|} = \frac{1}{|1-n^2|x_0||}= \left\{ \begin{array}{rcl}
    \frac{1}{1-n^2|x_0|} & \mbox{for}
    & n<\sqrt{\frac{1}{|x_0|} \\
    \frac{1}{n^2|x_0|-1} & \mbox{for}
    & n>\sqrt{\frac{1}{|x_0|}
    \end{array}\right [/tex]

    and

    [tex]\sum_{n=1}^{\infty}|f_n(x_0)| = \sum_{n=1}^{\left[\sqrt{1/|x_0|}\right]}\frac{1}{1-n^2|x_0|} + \sum_{n=\left[\sqrt{1/|x_0|}\right]+1}^{\infty}\frac{1}{n^2|x_0|-1}[/tex]

    I'm guessing this serie converges, but I'm having trouble finding a convergent serie to bound it with. The other convergence tests have failed and the use of the integral convergence criterion is forbiden. I know that if there is a serie to bound it with, it would be of the form

    [tex]\sum_{n=1}^{\infty}a_n = \sum_{n=1}^{\left[\sqrt{1/|x_0|}\right]}\frac{1}{1-n^2|x_0|} + \sum_{n=\left[\sqrt{1/|x_0|}\right]+1}^{\infty}b_n[/tex]

    with

    [tex]\frac{1}{n^2|x_0|-1} \leq b_n[/tex]

    for n > N.


    Edit:

    And if there exists such an N that also satisfies

    [tex]N\leq \left[\sqrt{1/|x_0|}\right][/tex]

    then according to Weirstrass M-test, the convergence is uniform.
     
    Last edited: Apr 4, 2005
  2. jcsd
  3. Apr 4, 2005 #2

    mathman

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    Because of all the singularities (0, -1/n2) in the interval, it can't converge uniformly.
     
  4. Apr 4, 2005 #3

    quasar987

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    That was not well said. What I meant to say is, does it converge pointwise and uniformly for the intervals in (-1,0) where the serie is defined. I.e. in the intervals

    [tex]\left(\frac{-1}{m^2} \ ,\frac{-1}{(m+1)^2}\right), & m \in \mathbb{N}[/tex]
     
  5. Apr 5, 2005 #4

    mathman

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    In the intervals of interest it converges pointwise, but not uniformly because of the blow ups at the end points of each interval.
     
  6. Apr 5, 2005 #5

    quasar987

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    On the basis of which theorem(s) are these statements made true? I would apreciate a quick answer because I need to hand out this question tomorrow!!

    Thanks!
     
    Last edited: Apr 5, 2005
  7. Apr 5, 2005 #6

    quasar987

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    By the way, I have found how to prove the pointwise convergence, I just don't know how to prove that it's not uniformly convergent on these intervals.
     
  8. Apr 6, 2005 #7

    mathman

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    I don't know what approach you are using to prove pointwise convergence. However, if you are using the old fashioned epsilon delta argument, you will see that there is a dependence on x when x is near a singular value.
     
  9. Apr 6, 2005 #8

    quasar987

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    I noticed that like 10 minutes before handing it out :biggrin:
     
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