Hi I'm kinda new here, but you may be my only hope, I've got 4 extremely hard problems. Here's one of them:
I've been trying to solve these for more than 8 hours now, but nothing seems to get me anywhere.
Thanks in advance!
Thanks a lot! Now that's a trick!
I don't want to be shameless, but I'm really hopeless with these problems.
These 2 are related in some way I think:
1. Prove that this is, or isn't convergent:
2. For what p, and q is this convergent, and for what p, and q is this absolute convergent?
Wait, what you needed is to prove that the integral is convergent? Then my trick is not the good one. Well, you still can split on and , but then this is the substitution that would be nice, since you end up with , and you only have to check that the function is integrable on .
To do this, notice that this function is continuous on , and as , because converges to 0 (and using a property of [tex]\ln[/Math] and asymptotic equivalence). Then, since converges, converges as well.
The trick I gave relates to the computation of the integral (once you know it exists). Using it, you get , and the substitution gives , and you deduce the value for your integral. But this assumes you proved its existence first.
I knew this had been done before. See here and go down to post #50:
http://www.mathhelpforum.com/math-he...tegrals-2.html
This link has many fun tough integrals