PreprintCC BY 4.0arXiv
On Sum Ranges for 3n-convergence
Abstract
The Riemann Rearrangement Theorem (RRT) tells us that commutativity of infinite series can differ from finite series. We wish to extend the notion of infinite series rearrangements to weaker forms of convergence, such as partial series convergence on every 2nd or 3rd index, called 2n- or 3n-convergence. In the case of 2n-convergence, it has been shown that the sum range, along with the standard cases in the RRT, can produce a shifted additive subgroup of reals. In this paper, we show cases where the 3n-sum range is still a subgroup and a case where the 3n-sum range cannot be a subgroup, proving the existence of more depth for 3n-convergence.
Related
BibTeX
@misc{martens2026ranges,
title = {{On Sum Ranges for $3n$-convergence}},
author = {Preston Martens},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01926},
eprint = {2609.01926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01926},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Sum Ranges for 3n-convergence
AU - Preston Martens
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01926
UR - https://arxiv.org/abs/2609.01926
AB - The Riemann Rearrangement Theorem (RRT) tells us that commutativity of infinite series can differ from finite series. We wish to extend the notion of infinite series rearrangements to weaker forms of convergence, such as partial series convergence on every 2nd or 3rd index, called 2n- or 3n-convergence. In the case of 2n-convergence, it has been shown that the sum range, along with the standard cases in the RRT, can produce a shifted additive subgroup of reals. In this paper, we show cases where the 3n-sum range is still a subgroup and a case where the 3n-sum range cannot be a subgroup, proving the existence of more depth for 3n-convergence.
ER -