Absolute Convergence

Let’s apply one of the tests from last time. Let $\alpha$ be a nondecreasing integrator on the ray $\left[a,\infty\right)$ , and $f$ be any function integrable with respect to $\alpha$ through the whole ray. Then if the improper integral $\int_a^\infty|f|d\alpha$ converges, then so does $\int_a^\infty f\,d\alpha$ .

To see this, notice that $-|f(x)|\leq f(x)\leq|f(x)|$ , and so $0\leq|f(x)|+f(x)\leq2|f(x)|$ . Then since $\int_a^\infty2|f|\,d\alpha$ converges we see that $\int_a^\infty|f|+f\,d\alpha$ converges. Subtracting off the integral of $|f|$ 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 $|f|$ converges like this, we say that the integral of $f$ 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 $f$ converges, but that of $|f|$ doesn’t, we say that the former is “conditionally convergent”.

3 Comments »

[…] also can import the notion of absolute convergence. We say that a series is absolutely convergent if the series is convergent (which implies that […]

Pingback by Convergence Tests for Infinite Series « The Unapologetic Mathematician | April 25, 2008 | Reply
[…] Ratio and Root Tests Now I want to bring out with two tests that will tell us about absolute convergence or (unconditional) divergence of an infinite series . As such they’ll tell us nothing about […]

Pingback by The Ratio and Root Tests « The Unapologetic Mathematician | May 5, 2008 | Reply
[…] integral, we have analogues of the direct comparison and limit comparison tests, and of the idea of absolute convergence. Each of these is exactly the same as before, replacing limits as approaches with limits as […]

Pingback by Improper Integrals II « The Unapologetic Mathematician | January 15, 2010 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

Recent Posts
Blogroll
Art
- Sketches of Topology
Astronomy
- Bad Astronomy
- Starts With a Bang
Computer Science
Education
Mathematics
Me
- DrMathochist
- The Unapologetic Programmer
Philosophy
Physics
Politics
Science

M	T	W	T	F	S	S
« Mar				May »
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30

The Unapologetic Mathematician

Mathematics for the interested outsider