Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in person or within a text, the discussion sort of ends after Riemann's theorem is given---quite content in proving to the student(s) or the reader that one shouldn't extend finite intuition to infinite settings without providing a proof.

I agree that this is an important moral to impart; however, I'm interested in something else:

Has the Riemann Rearrangement theorem been used as a computational aid to explicitly calculate a sum?

By this vague question, I specifically have in mind that piece of Riemann's Theorem that states an absolutely convergent series is commutatively convergent. So, to be slightly more narrow in scope:

Has there been a series $\sum a_k$ which is fairly easy to show absolutely converges; however, the sum itself was computed by a clever choice of bijection $\sigma:\mathbb{N}\rightarrow\mathbb{N}$ and by working with the partial sums of $\sum a_{\sigma(k)}$?

This is a rather vague question, and I don't expect it to have much of an absolute answer. But I'm interested in any variety of answers, and I'm sure they'd be demonstrative and helpful to future readers.

Solution 1:

Proffering an application of the rearrangement theorem. I'm not sure that this is exactly what you want, because there is no actual calculation of the sum in a closed form.

The existence of doubly periodic functions. That is, functions $f(z)$ that are meromorphic on the entire complex plane, and have two periods, say $\omega_1,\omega_2\in\Bbb{C}$ that are linearly independent over $\Bbb{R}$. In other words, we require that the identity $$ f(z+m\omega_1+n\omega_2)=f(z) $$ holds for all non-poles $z$ and all integers $m,n$.

Let us write $$\Lambda=\{m\omega_1+n\omega_2\mid m,n\in\Bbb{Z}\}.$$ Under the assumption that $\omega_1/\omega_2\notin\Bbb{R}$ we have that $\Lambda$ is a discrete subset of $\Bbb{C}$ that is also an additive free abelian group of rank two.

A construction idea is to start with a doubly infinite sum $$ f_{\Lambda}(z)=\sum_{\lambda\in\Lambda}\frac1{(z-\lambda)^3}. $$ If we fix an $\epsilon>0$ it is not difficult to show that this sum converges absolutely and uniformly in the set $U(\epsilon)=\Bbb{C}\setminus\bigcup_{\lambda\in\Lambda}B(\lambda,\epsilon),$ where $B(\lambda,\epsilon)$ is the ball of radius $\epsilon$ around the point $\lambda$. A key ingredient in the proof is that, when grouping the terms according to $\max\{|m|,|n|\}$ we get a series majorized by $\sum_k 1/k^2$.

Consequently we are allowed rearrange the (countably infinitely many) terms of the series $f_\Lambda(z)$ as we see fit.

As the point $\omega_1+\lambda$ ranges over the additive group $\Lambda$ while $\lambda$ does, the rearrangement theorem precisely tells us that $$f_{\Lambda}(z+\omega_1)=f(z)$$ for all $z\in U(\epsilon)$. Repeating the same argument shows that $\omega_2$ is also a period.

It follows from standard results (Weierstrass M-test and the like) that the function $f_\Lambda(z)$ is then holomorphic in $\Bbb{C}\setminus\Lambda$, and has a triple pole at all the points of the lattice $\Lambda$. The famous Weierstrass $\wp$-function can be constructed from here. Essentially the function $f_\Lambda(z)$ is the derivative $\wp_{\Lambda}'(z)$. These doubly periodic functions come in handy when working on (complex) elliptic curves and such. I recommend the first chapter of Apostol's Modular Functions and Dirichlet Series for more on the details, and books dedicated to elliptic curves for more.