Type R $\hat{I}»$-Permutation Approach to Velleman's Open Problem

Abstract

Previously, mathematicians Steven Krantz and Jeffery McNeal studied a type of positive numbers permutation called $\hat{I}»$-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 $\hat{I}»$-permutation), it is called a "conditionally divergent series". In 2006, another mathematician Daniel Velleman raised an open problem related to $\hat{I}»$-permutation: for a conditionally divergent series ${\sum }_{n=0}^{\infty }{a}_n,n\in \mathbb{N},a_n\in \mathbb{R}$, let $S=\{L\in \mathbb{R}:L={\sum }_{n=0}^{\infty }{a}_{\ddot{I}ƒ\left(n\right)}$ $\text{for some }\hat{I}»\text{-permutation }\ddot{I}ƒ\}$, can $S$ ever be something between $\varnothing$ and $\mathbb{R}$? This paper is devoted to partially answering this open problem by considering a subset of $\hat{I}»$-permutation constraint by how we can permute, named type R $\hat{I}»$-permutation. Then we answer the analogous question about a subset of S with respect to type R $\hat{I}»$-permutation, named $Z_R=\{L\in \mathbb{R}:L={\sum }_{n=0}^{\infty }{a}_{\ddot{I}ƒ\left(n\right)}$ $\text{for some type R }\hat{I}»\text{-permutation }\ddot{I}ƒ\}$. We show that $Z_R$ is either $\varnothing$, 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.