Currently taking integral calculus and I have a problem that I have no idea how to approach.
Given the integral from 1 to infinity of dx/(x^p *(ln(x))^a)
- For what values of p and a does this converge
Even if you cant get the full solution, any thoughts on how to approach it would be greatly appreciated.
Integrating by parts we obtain...
(1)
... and then with another integration by parts...
(2)
If now we proceed indefinitely we obtain...
(3)
At this point a careful investigation of (3) permits us to conclude that the [improper] integral...
(4)
... converges if...
a)
b) is a non positive integer...
If is ...
(5)
The convergence in the case but non integer can be easily verified by comparison...
Kind regards