Banishing divergence Part 1: Infinite numbers as the limit of sequences of real numbers

TL;DR

This paper introduces a non-Archimedean framework for infinite numbers, enabling the analysis of non-oscillating divergent sequences by defining classes of infinite numbers and their limits.

Contribution

It constructs a new ordered field of infinite numbers containing real, infinite, and infinitesimal elements, extending the analysis of divergent sequences.

Findings

Defines Archimedean classes of infinite numbers.

Introduces prototype sequences as asymptotes.

Establishes the set of ratios of limits as an ordered field.

Abstract

Sequences diverge either because they head off to infinity or because they oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers that is large enough to contain asymptotic limit points for non-oscillating sequences that head off to infinity. It begins by defining Archimedean classes of infinite numbers. Each class is denoted by a prototype sequence. These prototypes are used as asymptotes for determining leading term limits of sequences. By subtracting off leading term limits and repeating, limits are obtained for a subset of sequences called here ``smooth sequences". $\mathbb{I}_n$ is defined as the set of ratios of limits of smooth sequences. It is shown that $\mathbb{I}_n$ is an ordered field that includes real, infinite and infinitesimal numbers.