A Widely Applicable Nonstandard Test for Series Convergence


The use of infinitesimal methods (nonstandard analysis) in calculus can simplify computations, including the determination of convergence or divergence of a series. The level comparison test for series with nonnegative terms is an example. Featuring a computation that is similar in difficulty to the test for divergence, this test hinges on whether the reciprocal of the “omegath” term of the series lies in the “convergence zone” or in the “divergence zone.” In this talk the test is described, justified, and demonstrated. (The level comparison test is introduced in the textbook Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press, 2022.)