Integral Problems

Date: 9/14/95 at 7:15:15
From: Filiberto Strazzari
Subject: Generalized Integrals

Let's consider the function

                    1
     f(x) = -----------------
             x^h * (log x)^k

For what real values of h and k does

  infinite                      1
  /                           /
  | f(x) dx          and      | f(x) dx
  /                           /
   1                           0

converge or diverge?

Actually, I should add I'm in little hurry about this 
problem. Thank you very much for your effort.
 
                                Filiberto Strazzari
                           <def0721@iperbole.bologna.it>

Date: 10/11/95 at 17:17:39
From: Doctor Ethan
Subject: Re: Generalized Integrals

Well, here are my thoughts on your problem.

One way to phrase this question is "For what values of h and 
k does the bottom grow fast enough so that the integral 
converges?"

Let's look at integral one.  Both things on the bottom are 
positive for all of these values, so there is no fear that 
they will cancel each other out.

When h>1  1/x^h will converge by itself, so adding that extra 
stuff for any k will be okay.

So look at h<=1.  You may know that 1/x does not converge,
so one way to think about this is, "When is log[x]^k going 
to grow faster than x in the long run? "  We will show that 
the answer is that it never will.  

The notion of "grow faster" is very closely related to the 
derivative, so let us compare the derivative of the two 
functions.  The derivative of x is 1 and the derivative of 
log[x]^k is k * log[x]^(k-1) *1/x . But as x approaches 
infinity this approaches zero.  (You can prove this using 
L'Hopital's rule.)

So this tells that for the first one, when h<=1 this will not 
converge for any value of k.

Great! Now, on to the second one.

I will just give you a hint and let you see if you can figure 
it out.

We know that the polynomial part (i.e. x^h) will have finite 
value for h=0 or greater.  So for those you can just examine 
possible k values in a similar way to what we did earlier.

Now think about the values for h<0. These start to get 
interesting.  Think about them.

Good Luck...

- Doctor Ethan,  The Geometry Forum