# The rearrangement number, CUNY set theory seminar, November 2015

This will be a talk for the CUNY Set Theory Seminar on November 6, 2015.

The Riemann rearrangement theorem states that a convergent real series $\sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $\sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if the series is merely conditionally convergent, then one may find rearrangements for which the new sum $\sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent.  In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence.


## 3 thoughts on “The rearrangement number, CUNY set theory seminar, November 2015”

1. The paper is still in preparation, but we know much more than currently appears on the MathOverflow posts. Once we have a suitable draft, we’ll place it on the arxiv and also update the MO posts.