Testing for integral convergence

In summary, the conversation discusses how to test the convergence of the integral \int^{∞}_{-∞}\frac{dx}{\sqrt{x^{4}+1}} using the ratio test. It is shown that the integral converges if and only if the integral \int^{∞}_{-∞}\frac{dx}{x^{2}} converges, but it is not clear whether the original integral converges or diverges. It is noted that the function is not continuous, but it is almost continuous with a countable set of discontinuities. The conversation then moves on to discussing the p comparison test and finding a function to compare with for convergence. Ultimately, it is suggested to break the integral into intervals
  • #1
Bipolarity
776
2

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
 
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  • #2
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]
 
  • #3
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
 
  • #4
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.
 
  • #5
Zondrina said:
Break the integrand up and re-arrange your limits of integration.

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

BiP
 
  • #6
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?
 
  • #7
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
 
  • #8
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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  • #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
 
  • #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

1. What is the purpose of testing for integral convergence?

The purpose of testing for integral convergence is to determine whether a given integral converges or diverges. This is important in mathematical and scientific analysis to understand the behavior of functions and to make accurate predictions.

2. How do you test for integral convergence?

There are several methods for testing integral convergence, including the comparison test, the limit comparison test, the ratio test, and the integral test. These methods involve evaluating the behavior of the integrand and comparing it to known convergent or divergent functions.

3. What is the integral test?

The integral test is a method for testing the convergence of an infinite series by comparing it to a related improper integral. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

4. Can the integral test be used for all types of integrals?

No, the integral test can only be used for improper integrals, where the limits of integration are infinite or the integrand is unbounded.

5. What happens if the integral test is inconclusive?

If the integral test is inconclusive, other methods such as the comparison test or the ratio test can be used to determine convergence or divergence. In some cases, the integral test may need to be modified or combined with other methods to reach a conclusion.

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