Test the following improper integral for convergence:
The integral converges.
Proof. We can compute the value of the improper integral directly. First, we can evaluate the indefinite integral using integration by parts with
So, to evaluate the improper integral we take the limit,
Subsitute 1/x=t. Then limit comparison of the negated value of the integral with x^(-5/4).
The substitution for will not give the limit 0, since logarithm is not defined there – it is an indeterminate form
I believe you can rewrite
can be rewritten as
then we can just spam l’Hôpital and find that the limit is, in fact, equal to 0.
this an example in the section about infinite limit