Conditional Convergence and its pathologies

Definition 7.1   A series is said to be conditionally convergent if $\sum a_n$ converges but $\sum \vert a_n\vert$ does not converge.

The alternating harmonic series $${\sum \frac{(-1)^n}{n}}$$ is an example of this kind of behavior.

Conditionally convergent series will have a a subsequence of positive terms and a subsequence of negative terms which both give series which diverge. The individual terms $a_n\to 0$ since otherwise the series would be divergent. These can be used to produce a rearrangement of the series which converges to any value we wish. To make the limit L what you do is take positive terms until the first time that the sum exceeds L, then take negative terms until you first get a sum which is less than L. Each time you step across L you will be taking smaller steps so the resulting series will converge to L.

A similar argument will show that you can rearrange a conditionally convergent series to diverge as well.

Since the rearrangement of series is an infinite analog of the commutative law, we must consider this behavior pathological. Fortunately, inside the radius of convergence of a power series we get absolute convergence, which does not share this behavior.