Editing Alternating series (section)

== Rearrangements ==
For any series, we can create a new series by rearranging the order of summation. A series is [[Series (mathematics)#Unconditionally convergent series|unconditionally convergent]] if any rearrangement creates a series with the same convergence as the original series. [[Absolute convergence#Rearrangements and unconditional convergence|Absolutely convergent series are unconditionally convergent]]. But the [[Riemann series theorem]] states that conditionally convergent series can be rearranged to create arbitrary convergence.<ref>{{cite journal |last1=Mallik |first1=AK |year=2007 |title=Curious Consequences of Simple Sequences |journal=Resonance |volume=12 |issue=1 |pages=23–37 |doi=10.1007/s12045-007-0004-7|s2cid=122327461 }}</ref> [[Agnew's theorem]] describes rearrangements that preserve convergence for all convergent series. The general principle is that addition of infinite sums is only commutative for absolutely convergent series.

For example, one false proof that 1=0 exploits the failure of associativity for infinite sums.

As another example, by [[Mercator series]]
<math display="block">\ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.</math>

But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for <math display="inline">\tfrac 1 2 \ln(2)</math>:
<math display="block">\begin{align}
& {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\[8pt]
& = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\[8pt]
& = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2).
\end{align}</math>