Little theorem - Convergence of improper integral

In summary, the conversation discusses the convergence of integrals and the use of the Dirichlet Convergence Test with the Fundamental Theorem of Calculus. It is mentioned that the convergence of an integral is equivalent to the convergence of its absolute value, and that if f(x)/x <= f(x) for x >= 1, then the convergence of int(f(x)) implies the convergence of int(f(x)/x). However, it is noted that this is only true if the integral from 1 to infinity is absolutely convergent. A specific example is given to illustrate this. The conversation ends with an individual expressing interest in seeing the proof for the theorem.
  • #1
estro
241
0
[PLAIN]http://estro.uuuq.com/_proof.jpg

I think I miss something...
 
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  • #2
What I did is wrong, however I figured out what was wrong and no further help is needed.
If someone is interested I'll post the right proof. (I've used "Dirichlet Convergence Test" with "Fundamental Theorem of Calculus").
 
  • #3
Hi,
I must be mistaken, but I don't know where. Could you please correct me?:

* int(f(x), x=1..infinity) converges is equivalent to int(abs(f(x)),x=1..infinity) coonverges;
*for x>=1, f(x)/x<=f(x).
*then, int(f(x)) converges implies int(f(x)/x) converges.
??
 
  • #4
This is true only if, integral (f(x)dx) from 1 to infinity, is absolutely convergent.
For example, integral (cosx/x) from 1 to infinity, is convergent but, integral abs(cosx/x) from 1 to infinity diverges.
 
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  • #5
Thanks estro,
I would be very interested in your proof, if you don't mind then...
 
  • #6
penguin007 said:
Thanks estro,
I would be very interested in your proof, if you don't mind then...

[PLAIN]http://estro.uuuq.com/_proof22.jpg
 
Last edited by a moderator:
  • #7
thank you for your proof estro.
 

Related to Little theorem - Convergence of improper integral

1. What is Little's theorem?

Little's theorem, also known as the First Law of Convergence, is a theorem in mathematics that states that a sequence of numbers converges if and only if it is a Cauchy sequence. This theorem is used to prove the convergence of improper integrals.

2. What is an improper integral?

An improper integral is an integral where either the upper or lower limit of the integral is infinite, or the integrand is not defined at some point within the interval of integration. These types of integrals do not have a finite solution and thus require special methods, such as Little's theorem, to determine their convergence.

3. How does Little's theorem help determine the convergence of improper integrals?

Little's theorem provides a necessary and sufficient condition for the convergence of a sequence, which can then be applied to improper integrals. By showing that a sequence of numbers is a Cauchy sequence, we can prove that the corresponding improper integral also converges.

4. Are there any limitations to Little's theorem?

Yes, Little's theorem only applies to sequences of numbers and cannot be used to determine the convergence of other types of mathematical objects. Additionally, it only provides a sufficient condition for convergence, meaning that a sequence may still converge even if it does not satisfy the conditions of Little's theorem.

5. How is Little's theorem used in real-world applications?

Little's theorem is used in a variety of fields, including engineering, physics, and economics, to determine the convergence of various mathematical models and equations. It is particularly useful in systems analysis and queueing theory, where it is used to analyze the behavior and performance of queues and networks.

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