## Real Analysis Exchange

### The Descriptive Complexity of Series Rearrangements

Michael P. Cohen

#### Abstract

We consider the descriptive complexity of some subsets of the infinite permutation group $S_\infty$ which arise naturally from the classical series rearrangement theorems of Riemann, Levy, and Steinitz. In particular, given some fixed conditionally convergent series of vectors in Euclidean space $\mathbb{R}^d$, we study the set of permutations which make the series diverge, as well as the set of permutations which make the series diverge properly. We show that both collections are $\boldsymbol\Sigma^0_3$-complete in $S_\infty$, regardless of the particular choice of series.

#### Article information

Source
Real Anal. Exchange, Volume 38, Number 2 (2012), 337-352.

Dates
First available in Project Euclid: 27 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.rae/1403894896

Mathematical Reviews number (MathSciNet)
MR3261881

Zentralblatt MATH identifier
1304.40001

#### Citation

Cohen, Michael P. The Descriptive Complexity of Series Rearrangements. Real Anal. Exchange 38 (2012), no. 2, 337--352. https://projecteuclid.org/euclid.rae/1403894896