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

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.