Absolute Convergence
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”.
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