Type R $\lambda$-Permutation Approach to Velleman's Open Problem

TL;DR

This paper investigates a specific class of permutations called type R λ-permutations to address Velleman's open problem on the possible sets of sums of conditionally divergent series after permutation, showing these sets are either empty, singleton, or all real numbers.

Contribution

It introduces type R λ-permutations and characterizes the possible sum sets, providing conditions under which these sets are singleton or the entire real line.

Findings

The set of sums under type R λ-permutations is either empty, a singleton, or all real numbers.

Sufficient conditions are given for the sum set to be a singleton or all real numbers.

A 'substantial property' on series determines the structure of the sum set.

Abstract

Previously, mathematicians Steven Krantz and Jeffery McNeal studied a type of positive numbers permutation called $\lambda$-permutation. This type of permutation, when applied to the index of terms of a series, is defined to be both convergence-preserving and "fixing" at least one divergent series, that is, rearranging the terms of any convergent series will result in a convergent series, while rearranging the terms of some divergent series will result in a convergent series. In general, if a divergent series can be fixed to converge in some way (it does not need to be by $\lambda$-permutation), it is called a "conditionally divergent series". In 2006, another mathematician Daniel Velleman raised an open problem related to $\lambda$-permutation: for a conditionally divergent series $\sum_{n=0}^{\infty}a_n,n\in \mathbb{N},a_n\in \mathbb{R}$, let $S=\{L \in \mathbb{R} \colon L = \sum_{n=0}^{\infty}{a_{\sigma\left(n\right)}}$ $\text{for some } \lambda\text{-permutation } \sigma\}$, can $S$ ever be something between $\emptyset$ and $\mathbb{R}$? This paper is devoted to partially answering this open problem by considering a subset of $\lambda$-permutation constraint by how we can permute, named type R $\lambda$-permutation. Then we answer the analogous question about a subset of S with respect to type R $\lambda$-permutation, named $Z_{R}=\{L \in \mathbb{R} \colon L = \sum_{n=0}^{\infty}{a_{\sigma\left(n\right)}}$ $\text{for some type R } \lambda \text{-permutation } \sigma\}$. We show that $Z_R$ is either $\emptyset$, a singleton or $\mathbb{R}$. We also provide sufficient conditions on the conditionally divergent series $\sum_{n=0}^{\infty}a_n$ for $Z_R$ to be a singleton or $\mathbb{R}$, by introducing a "substantial property" on the series.