Alternating Series - Rearrangements

Rearrangements

For any series, we can create a new series by rearranging the order of summation. A series is unconditionally convergent if any rearrangement creates a series with the same convergence as the original series. Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence. The general principle is that addition of infinite sums is only associative for absolutely convergent series.

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

As another example, we know that .

But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for :

$\begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4}+\left(\frac{1}{3}-\frac{1}{6}\right)-\frac{1}{8}+\left(\frac{1}{5}-\frac{1}{10}\right)-\frac{1}{12}+\cdots \\ & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-\frac{1}{12}+\cdots \\ & = \frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots\right)= \frac{1}{2} \ln(2) \end{align}$

Another valid example of alternating series is the following

$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k+1}}=1-\frac{1}{\sqrt{2}} +\frac{1}{\sqrt{3}} -\frac{1}{\sqrt{4}} +\frac{1}{\sqrt{5}} \cdots=-(\sqrt{2} -1)\zeta(\frac{1}{2})\approx0.6048986434...$

Read more about this topic: Alternating Series