Let’s apply one of the tests from last time. Let be a nondecreasing integrator on the ray , and be any function integrable with respect to through the whole ray. Then if the improper integral converges, then so does .
To see this, notice that , and so . Then since converges we see that converges. Subtracting off the integral of we get our result. (Technically to do this, we need to extend the linearity properties of Riemann-Stieltjes integrals to improper integrals, but this is straightforward).
When the integral of converges like this, we say that the integral of is “absolutely convergent”. The above theorem shows us that absolute convergence implies convergence, but it doesn’t necessarily hold the other way around. If the integral of converges, but that of doesn’t, we say that the former is “conditionally convergent”.