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.