Convergens of odd integral

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In summary, the conversation is about proving that f(x) tends to 0 as x approaches infinity and whether it is convergence or simply existence of the limit. The question also involves the use of an integral.
  • #1
Math_Frank
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Homework Statement



Given the odd integral

[tex]\int_{a}^{b} f(x) dx[/tex] How do I prove that

f(x) -> 0 for [tex]x \to \infty[/tex]??

The Attempt at a Solution



Is it? For the above to be true, then there exist an [tex]\epsilon > 0[/tex] such that

[tex]|\int_{a}^{b} f(x) dx-0| \leq \epsilon[/tex]?

I am stuck here!

Am I going the right way?

Sincerely
Frank
 
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  • #2
What you've written doesn't really make sense. What is this question from and about?
 
  • #3
NateTG said:
What you've written doesn't really make sense. What is this question from and about?

The Question is

Given the integeral

[tex]f(t) = \int_{t}^{2t} e^{-x^2} dx [/tex] then prove that if [tex]f(x) \to 0[/tex] then

[tex]n \to \infty[/tex]

Isn't that convergens or it simply existence of the limit?
 
  • #4
Math_Frank said:
The Question is

Given the integeral

[tex]f(t) = \int_{t}^{2t} e^{-x^2} dx [/tex] then prove that if [tex]f(x) \to 0[/tex] then

[tex]n \to \infty[/tex]

Isn't that convergens or it simply existence of the limit?

Where does [itex]n[/itex] come from?

Do you mean "[itex]\lim_{x \rightarrow \infty} f(x)=0[/itex]" when you write "[itex]f(x) \to 0[/itex]"
 
  • #5
NateTG said:
Where does [itex]n[/itex] come from?

Do you mean "[itex]\lim_{x \rightarrow \infty} f(x)=0[/itex]" when you write "[itex]f(x) \to 0[/itex]"

Yes.
 
  • #6
You need to show both existence and convergence of the limit.
 

1. What does "convergence of odd integral" mean?

The convergence of odd integral refers to the behavior of an integral when the limits of integration are symmetric about the origin (i.e. one limit is positive and the other is negative). In this case, the integral may have special convergence properties that differ from integrals with non-symmetric limits.

2. How is the convergence of odd integral different from that of even integral?

The convergence of odd integral differs from that of even integral because the symmetry of the limits of integration in odd integrals can result in the integral converging even when the integrand function does not approach a finite limit as the limits of integration approach infinity.

3. What are some examples of functions with odd integrals?

Some examples of functions with odd integrals include sine, cosine, tangent, cotangent, and other trigonometric functions, as well as rational functions with odd powers in the denominator.

4. How can I determine the convergence of an odd integral?

To determine the convergence of an odd integral, you can use techniques such as the comparison test, limit comparison test, or integration by parts. It is also helpful to graph the integrand function and observe its behavior near the limits of integration.

5. What are some applications of the convergence of odd integral?

The convergence of odd integral is important in various areas of mathematics and physics, such as Fourier series, Laplace transforms, and calculating the work done by a force over a displacement. It can also be used to evaluate certain improper integrals that would otherwise be divergent.

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