Szego Seminar: Rearranging the Riemann Rearrangement Theorem

Abstract: The Riemann Rearrangement Theorem tells us that, under a suitable permutation of its terms, a conditionally convergent series may be made to converge to any real number, or to diverge. What if instead of fixing a series and looking at different permutations, one were to consider each possible permutation of the natural numbers, and ask what it does to the class of conditionally convergent series? Doing this, one will stumble across the surprising fact that there exist permutations which not only preserve all convergence series, but turn some divergent series into convergent series; that is, they "improve" on the set of convergent series. In this talk, I will construct such permutations, and discuss some of their interesting properties.

Host: Pooja Joshi