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

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