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.
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.
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.