# Homework Help: Little theorem - Convergence of improper integral

1. May 6, 2010

### estro

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

I think I miss something...

Last edited by a moderator: May 4, 2017
2. May 6, 2010

### 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").

3. May 6, 2010

### 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.
??

4. May 6, 2010

### 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.

Last edited: May 7, 2010
5. May 7, 2010

### penguin007

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

6. May 7, 2010

### estro

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

Last edited by a moderator: May 4, 2017
7. May 9, 2010

### penguin007

thank you for your proof estro.