We often use integrals of the functions , for various values of p, to help determine whether other integrals converge or diverge.
You already did the work to show this, so we'll just summarize the results. Assuming p is greater than 0 (because otherwise the exponents do weird things),
- converges if p < 1 and diverges otherwise.
- converges if p > 1 and diverges otherwise.
This is often called the p-test for improper integrals.