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Testing for integral convergence

  1. Sep 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Test the following integral for convergence:
    [tex] \int^{∞}_{-∞}\frac{dx}{\sqrt{x^{4}+1}} [/tex]

    2. Relevant equations



    3. The attempt at a solution
    I was able to use the ratio test to show that the integral converges if and only if [itex] \int^{∞}_{-∞}\frac{dx}{x^{2}} [/itex] converges, but I haven't been able to show whether this particular integral converges or diverges.
    This integral is not continuous, but it is almost continuous since the discontinuity is a countable set.

    NOTE: The integral on the bottom actually diverges, and yet the original integral converges!!! But HOW?!!!


    BiP
     
    Last edited: Sep 17, 2012
  2. jcsd
  3. Sep 17, 2012 #2

    Zondrina

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    Do you remember the p comparison test for :

    [tex]\int_{-∞}^{∞} \frac{1}{x^p}[/tex]
     
  4. Sep 17, 2012 #3
    Yup, but note that the function is discontinuous in the interval.

    BiP
     
  5. Sep 17, 2012 #4

    Zondrina

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    Break the integrand up and re-arrange your limits of integration.
     
  6. Sep 17, 2012 #5
    Ok, I edited my original post. Please re-read it.

    BiP
     
  7. Sep 17, 2012 #6

    Zondrina

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    Are you sure 1/x^2 converges? I just ran through this really quickly and I'm getting that this definitely diverges?
     
  8. Sep 17, 2012 #7
    Yes, which is why I edited on the NOTE to say that it does indeed diverge.
    The original problem however happens to converge, even though the ratio test implied they would behave the same way.

    BiP
     
  9. Sep 17, 2012 #8

    Dick

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    1/x^2 converges on the intervals [1,infinity) and (-infinity,-1]. You'll have to use a different function on the interval [-1,1] for a comparison test. Then add the three together.
     
    Last edited: Sep 17, 2012
  10. Sep 17, 2012 #9
    I've been searching for a while but can't find a function to compare with for convergence. What do you guys suggest?

    BiP
     
  11. Sep 17, 2012 #10

    Dick

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    I hope you've broken it up into intervals. If so, then for [-1,1] how about just using 1? That's bigger than your integrand.
     
  12. Sep 17, 2012 #11
    It converges! Thank you!

    BiP
     
  13. Sep 17, 2012 #12

    Zondrina

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    Assume that your integral dx/(x^4+1) is denoted by f(x). Assume the function you're comparing it to 1/x^2 is g(x).

    Now I'm not sure you're allowed to do this, but find :

    [tex]lim_{x} \rightarrow ∞ \frac{f(x)}{g(x)}[/tex]

    This will tell you whether both functions converge or both functions diverge. If your limit is positive and finite, but less than infinity, then your original series will converge.
     
  14. Sep 17, 2012 #13

    Dick

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    There are premises that need to be satisfied to use that test. For one thing 1/x^2 needs to be continuous on the domain of integration. It's not.
     
  15. Sep 17, 2012 #14

    Zondrina

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    Oh really? I completely forgot about that apparently. I got this theorem from here though, maybe he didn't mention it explicitly?

    http://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx
     
  16. Sep 17, 2012 #15
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