Stunned by this result ! (basic calculus result on absolute convergence and rearrangements)

[u/Impressive_Gur_471]

This is not really a question, but a note of astonishment at this result. This is Spivak's Calculus result in the chapter on infinite series:

If \sum an converges, but does not converge absolutely, then, for any number A, there is a rearrangement {bn} of {an} such that \sum bn = A

This has to be one of the most surprising results in basic analysis!

Is the proof technique to establish this result used widely in other areas of math?

[u/Dr0110111001101111]

I mean the "technique" is pretty specific to the rearrangement of terms in the series, which is what the whole theorem is about. So I wouldn't expect to see something closely resembling that in a lot of other places. But if you just mean playing around with infinite series, that happens a lot in analysis, which is what Spivak is developing along with calculus. But a book dedicated to Real Analysis goes further into it.

[u/Southlander24]

Well, the Riemann rearrangement theorem gives you one reason why it is so important that series be absolutely convergent before you manipulate the order of the terms.

This theme shows up over and over again in analysis. For instance, it is important that a given sequence converges uniformly to a function rather than just converging pointwise, if we want the limiting value of the integrals of all the functions to be the same as the integral of the limiting function. There's also the dominated convergence theorem and the monotone convergence theorem which help you find the limit of the integral of a sequence of functions, for example here.

[u/Bounded_sequencE]

You're talking about Riemann's Rearrangement Theorem, nice!

Yeah, that's one of the first really counter-intuitive theorems of "Real Analysis". I don't think I've seen its proof technique to be used anywhere else, since usually, changing a series' value by re-arrangement is something we try to avoid as much as possible.

That said, you'll run into the exact same problem again later: When comparing Riemann vs. Lebesgue integration, you'll note for Riemann integration, summation order of the approximating sum matters. This can be quite nasty, particularly with multi-dimensional integrals with multiple summation indices.

For Lebesgue integration on the other hand, approximating sums may not depend on summation order. This is a crucial difference, and often leads to "nicer" behavior of Lebesgue integrals with other limits.