Absolute Convergence vs. Conditional Convergence
As with most things in math, there are a few things that just can't fit nicely into the standard size boxes we try to to put them in. In the case of a few series, we need a Pandora's box that can open from 2 different ends.
Take a better look at what kind of beasts we're dealing with that we need a special box to constrain them. We can draw rectangles for series with negative terms the same way we drew left- and right-hand sums for integrals of functions with negative values. If a term is negative, we stick its rectangle under the horizontal axis.
Suppose is a series whose terms may be positive and/or negative. We can make another series by taking the absolute value of each term:
Visually, if the rectangles for look like this:
then to get the rectangles for we take any rectangles that were below the horizontal axis and flip them to lie on top of it instead:
If the original series converges but the series of absolute values doesn't, we say the original series converges conditionally.
These series are the Dr. Jekyll and Mr. Hyde of convergence. They are well-behaved on their own, but when you mix in a little potion with an absolute value symbol, they are out of control and diverge. If we try to put them in our Pandora's box, sometimes it's safe to open the box, and sometimes it's not.
Here's what conditional convergence looks like. Suppose has both positive and negative terms, and converges conditionally.
Since the limit of the partial sums converges, the positive and negative terms mostly cancel each other out as we go along:
Since converges conditionally, the series doesn't converge. This means the rectangles for cover an infinite area:
If the series of absolute values converges we say the original series converges absolutely. If the series of absolute values converges, it conveniently forces the original series to converge also. Put another way, if Mr. Hyde is well-behaved, so is Dr. Jekyll. The box is safe to open from either side.
Here's what absolute convergence looks like.
If converges, the region covered by all its rectangles is finite.
Any sub-region of a finite region is finite. That means the region covered by rectangles corresponding to positive terms of is finite.
And the region covered by rectangles corresponding to negative terms of is finite too.
The weighted area covered by the rectangles for must be the area above the axis minus the area below the axis:
This value is one finite number minus another, so it's finite, which means is finite. The series converges.
Practically speaking, this means if we can show converges, then we know that converges.
The alternating harmonic series converges conditionally but not absolutely, because
The geometric series converges absolutely because
converges, being a geometric series with |r| < 1.