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arXiv · 1108.5081

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

David Alan Paterson
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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.

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David Alan Paterson. 2011-08-23. Banishing divergence Part 1: Infinite numbers as the limit of sequences of real numbers. https://arxiv.org/abs/1108.5081

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