  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.
