Little theorem - Convergence of improper integral

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Homework Help Overview

The discussion revolves around the convergence of improper integrals, specifically related to the Little theorem and the conditions under which certain integrals converge or diverge. Participants are exploring the implications of absolute convergence and the use of convergence tests.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between the convergence of an integral and its absolute convergence, questioning the validity of certain assumptions and theorems. There are attempts to clarify the conditions under which specific integrals converge, with references to the Dirichlet Convergence Test and the Fundamental Theorem of Calculus.

Discussion Status

Some participants have acknowledged errors in their reasoning and have indicated a willingness to share corrected proofs. There is interest in further exploration of the topic, with multiple interpretations being considered regarding the convergence criteria.

Contextual Notes

Participants note that the discussion hinges on the distinction between absolute convergence and conditional convergence, with specific examples provided to illustrate these concepts. There is a mention of homework constraints that may limit the depth of exploration.

estro
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[PLAIN]http://estro.uuuq.com/_proof.jpg

I think I miss something...
 
Last edited by a moderator:
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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").
 
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.
??
 
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.
 
Last edited:
Thanks estro,
I would be very interested in your proof, if you don't mind then...
 
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:
thank you for your proof estro.
 

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