Proving Limits of Infinite Integrals with LaTeX

In summary, the conversation discusses the proof that if the integral of a function exists over the entire real line, then the limit of the integral from -N to N as N approaches infinity also exists and is equal to the original integral. The conversation also mentions the existence of the limits of the integrals from -N to N+1 and from -N^2 to N as N approaches infinity. The approach involves splitting up the integrals and using the triangle inequality to prove the result. However, the precise definition of the integral and whether it is the Riemann or Lebesgue integral is not specified, so it is important to clarify these details before proceeding with the proof.
  • #1
mscbuck
18
0

Homework Statement


Prove that if [tex] \int_{-\infty}^{+\infty} f [/tex] exists, then [tex]\lim_{N\rightarrow \infty[/tex] of [tex] {\int_{-N}^{N} f} [/tex] exists and is equal to the first equation.

Show moreover, that [tex]\lim_{N\rightarrow \infty[/tex] of [tex] {\int_{-N}^{N+1} f} [/tex] and [tex]\lim_{N\rightarrow \infty[/tex] of [tex] {\int_{-N^2}^{N} f} [/tex] both existThe attempt at a solution

It's taking me a really long time to write this out in LaTeX and it honestly looks worse than words when I finish it because I'm trying to learn it, so for now I"ll type in words.
My first step was split up the integrals. I have:

| Integral from 0->Inf of f MINUS Integral from 0->M of f | < 1/2E , and the correlating one for -Infinity to 0.

I then assumed an h(n) > M and a g(N) < -M for all N to come up with a generalization, and set up a large inequality and from that I believe by the triangle inequality I prove my result. But I"m unsure so if hopefully someone can check, that'd be great!

Thanks!
 
Last edited:
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  • #2
Because this is an area where different authors choose to set up their definitions in widely differing ways, it would be helpful if you gave your precise definition for [tex]\int_{-\infty}^{+\infty} f[/tex] (and for [tex]\int_0^\infty f[/tex] if that's involved in the previous definition). Also indicate whether this is the Riemann or Lebesgue integral. That way we can avoid leading you down a path that doesn't work with your exact set of assumptions.
 

Related to Proving Limits of Infinite Integrals with LaTeX

1. How do you use LaTeX to prove the limits of infinite integrals?

To use LaTeX to prove the limits of infinite integrals, you will first need to define the integral using the \int command. Then, you can use the \lim command to indicate that you are taking the limit of the integral as the variable approaches infinity. Finally, use the \rightarrow command to specify the value that the integral approaches as the limit.

2. What is the purpose of proving the limits of infinite integrals?

The purpose of proving the limits of infinite integrals is to determine the behavior of the integral as the variable approaches infinity. This can provide important information about the overall behavior of a function and help make predictions about its behavior on a larger scale.

3. Are there any specific syntax rules for using LaTeX to prove limits of infinite integrals?

Yes, there are some specific syntax rules that should be followed when using LaTeX to prove the limits of infinite integrals. For example, the \int command should always be followed by the limits of integration, and the \lim command should be followed by the variable and value of the limit. It is also important to use proper spacing and parentheses to ensure accuracy.

4. Can LaTeX be used to prove limits of infinite integrals for any type of function?

Yes, LaTeX can be used to prove limits of infinite integrals for any type of function, as long as the function is properly defined and the limits of integration are within the function's domain. However, some more complex functions may require additional steps or techniques to prove the limits.

5. Are there any alternatives to using LaTeX for proving limits of infinite integrals?

Yes, there are alternative methods for proving the limits of infinite integrals, such as using mathematical software or graphing calculators. However, LaTeX is a commonly used and efficient tool for displaying mathematical equations and proving limits, making it a popular choice among scientists and mathematicians.

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