Proving Convergence of f(x) Integrals in [a,∞)

[no_alone]

Homework Statement

f(x) positive and continuous in [a,$$\infty$$)
prove of disprove:
if $$\int_a^\infty f(x)dx$$ converge there is a 0<c<1 so that
$$\int_a^\infty f(x)^p dx$$ converge for every c$$\leq$$p$$\leq$$1

Homework Equations

every thing in calculus 1+2 no two vars ..

The Attempt at a Solution

Well I think that there is ..
But i have no clue on how to solve it...
Thank you.

 

[Dick]

I don't think it's true. Try find an example of an f(x) that just barely converges and f(x)^p diverges for all 0<p<1.

 

[no_alone]

Ok
what about
$$\frac{1}{xln^2(x)}$$
for every p<1
$$\frac{1}{x^pln^{2/p}(x)} > \frac{1}{x}$$

But i always have problem proving that for every p an z lim(x^p) > lim(ln^z(x))
Can you help me with this?

 

[rsa58]

notice that you function is undefined at x=0 and the inequality you propose is incorrect for 0<x<1

 

[rsa58]

sorry it is completely incorrect

 

[rsa58]

i have a feeling it is true since f(x)^p depends continously on p. are you sure this is all the info?

 

[no_alone]

notice that you function is undefined at x=0 and the inequality you propose is incorrect for 0<x<1

yes but I don't care about when 0<x<1 because its in a half closed section when f(X) is defined and continuous ,
I only look at infinity, and in infinity ,

for every p<1


$$\frac{1}{x^pln^{2/p}(x)} > \frac{1}{x}$$

 

[rsa58]

incorrect try x=3. the inequality is actually backwards. plus your function cannot include p. the result must hold for ALL p. since f is positive you can use the ration

 

[Dick]

Ok
what about
$$\frac{1}{xln^2(x)}$$
for every p<1
$$\frac{1}{x^pln^{2/p}(x)} > \frac{1}{x}$$

But i always have problem proving that for every p an z lim(x^p) > lim(ln^z(x))
Can you help me with this?

That's a great choice! It would be enough to show that in the limit ln(x)<x^p for all p>0, wouldn't it? Just use l'Hopital on the limit of x^p/ln(x).

 

[no_alone]

Thank you very much dick..
It's not really homework question,
Its a example test question .
This question really on my mind because it was really against my intuition ,
And my usually my intuition is true ,(my problem is proving it)
Can you supply another f(X) that is good?,
Thank you.

 

[Dick]

You're welcome. 'Intuition' can go a little wrong when you are dealing with an 'infinite' part in the problem. Guess I don't have a ready stock of problems. You might scan through other forum posts looking for interesting problems. There are some in there. And feel free to help the poster if you've got a good idea on how to solve it.

 

[no_alone]

You're welcome. 'Intuition' can go a little wrong when you are dealing with an 'infinite' part in the problem. Guess I don't have a ready stock of problems. You might scan through other forum posts looking for interesting problems. There are some in there. And feel free to help the poster if you've got a good idea on how to solve it.

Sorry dick but I think you misunderstood me,
I asked if you can give me another counter example beside this one I supplied?
Thank you.

 

[Dick]

Sorry dick but I think you misunderstood me,
I asked if you can give me another counter example beside this one I supplied?
Thank you.

Nothing pops into my head. There's 1/(log(x)^n*x) for n>1. But that's just a variation on the same theme.