Little theorem - Convergence of improper integral

[estro]

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

I think I miss something...

 

[estro]

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").

 

[penguin007]

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.
??

 

[estro]

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.

 

[penguin007]

Thanks estro,
I would be very interested in your proof, if you don't mind then...

 

[estro]

Thanks estro,
I would be very interested in your proof, if you don't mind then...

[PLAIN]http://estro.uuuq.com/_proof22.jpg

 

[penguin007]

thank you for your proof estro.