Convergence and divergence testing theory and applications by Integration at a point

TL;DR

This paper introduces a new integration method called convergence sums, which unifies and extends convergence tests for sums and integrals using a novel approach based on integration at a point and non-reversible arithmetic.

Contribution

It develops a comprehensive theory for convergence and divergence testing by extending du Bois-Reymond's theory with gossamer numbers, reforming known tests and connecting integration with series.

Findings

Reformulation of convergence tests and arrangement theorems.

Introduction of the boundary test as a universal convergence/divergence criterion.

Connection of integration and series through separation of finite and infinite domains.

Abstract

Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is continually increasing. Apply the fundamental theorem of calculus. The integral is a difference of the function integrated at the end points. If one of these point integrals is much-greater-than the other in magnitude delete it by non-reversible arithmetic. We call this type of integration "convergence sums" because our primary application is a theory for the determination of convergence and divergence of sums and integrals. The theory is far-reaching. It reforms known convergence tests and arrangement theorems, and it connects integration and series switching between the different forms. By separating the finite and infinite domains, the mathematics is more naturally considered, and is a problem reduction. In this endeavour we rediscover and reform the "boundary test" which we believe to be the boundary between convergence and divergence: the boundary is represented as an infinite class of generalized p-series functions. All this is derived from extending du Bois-Reymond's theory with gossamer numbers and function comparison algebra.