I know of several convergence/ divergence tests for infinite sums, but strangely the only tests I know for improper integrals (which seemingly are just continuous versions of infinite sums) are the comparison test and the p-test. Can we apply all of the same tests we use to determine convergence of infinite sums to improper integrals? If not, why?
In particular, I'm looking at an annoyingly difficult improper integral at the moment and I'm noticing that if the $n$th term test applies to improper integrals as well as infinite series, it'll diverge. So can I apply this test? If not, why?