Hyperreal Numbers for Infinite Divergent Series

Jonathan Bartlett; Logan Gaastra; David Nemati

doi:10.33014/issn.2640-5652.2.1.bartlett-et-al.1

Jonathan Bartlett
Logan Gaastra
David Nemati

DOI: https://doi.org/10.33014/issn.2640-5652.2.1.bartlett-et-al.1

Keywords: infinite series, abel summation, cesaro summation, hyperreal, number systems, discrete calculus, infinite, infinity, Grandi's series

Abstract

Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them. The infinities that resulted from divergent series led to contradictions within the real number system, but these contradictions are largely alleviated with the hyperreal number system. Hyperreal numbers provide a framework for dealing with divergent series in a more comprehensive and tractable way.

An unreviewed update/correction to Section 11.2 of this paper is available here.