Testing for integral convergence

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Homework Help Overview

The discussion revolves around testing the convergence of the integral \(\int^{∞}_{-∞}\frac{dx}{\sqrt{x^{4}+1}}\). Participants explore the relationship between this integral and the integral \(\int^{∞}_{-∞}\frac{dx}{x^{2}}\), noting that the latter diverges while the former converges, which raises questions about the underlying assumptions and methods used in convergence testing.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the use of the ratio test and the p-comparison test to analyze convergence. There are attempts to break the integral into intervals and consider different functions for comparison, particularly in the context of discontinuities and the behavior of the integrand.

Discussion Status

The discussion is ongoing, with participants providing guidance on breaking the integral into intervals and suggesting potential comparison functions. There is recognition of the need to address the discontinuity of the integrand and the implications for convergence tests.

Contextual Notes

Participants note that the function \(\frac{1}{x^2}\) is not continuous over the entire domain of integration, which raises questions about the validity of certain convergence tests. There is also mention of the differences between tests for series and integrals, highlighting potential misunderstandings in applying these concepts.

Bipolarity
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Homework Statement



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

Homework Equations


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:
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Bipolarity said:

Homework Statement



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

Homework Equations





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.

BiP

Do you remember the p comparison test for :

[tex]\int_{-∞}^{∞} \frac{1}{x^p}[/tex]
 
Zondrina said:
Do you remember the p comparison test for :

[tex]\int_{-∞}^{∞} \frac{1}{x^p}[/tex]

Yup, but note that the function is discontinuous in the interval.

BiP
 
Bipolarity said:
Yup, but note that the function is discontinuous in the interval.

BiP

Break the integrand up and re-arrange your limits of integration.
 
Zondrina said:
Break the integrand up and re-arrange your limits of integration.

Ok, I edited my original post. Please re-read it.

BiP
 
Bipolarity said:
Ok, I edited my original post. Please re-read it.

BiP

Are you sure 1/x^2 converges? I just ran through this really quickly and I'm getting that this definitely diverges?
 
Zondrina said:
Are you sure 1/x^2 converges? I just ran through this really quickly and I'm getting that this definitely diverges?

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
 
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.
 
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I've been searching for a while but can't find a function to compare with for convergence. What do you guys suggest?

BiP
 
  • #10
Bipolarity said:
I've been searching for a while but can't find a function to compare with for convergence. What do you guys suggest?

BiP

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.
 
  • #11
Dick said:
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.

It converges! Thank you!

BiP
 
  • #12
Bipolarity said:
I've been searching for a while but can't find a function to compare with for convergence. What do you guys suggest?

BiP

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.
 
  • #13
Zondrina said:
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.

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.
 
  • #14
Dick said:
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.

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
 
  • #15

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