The Case for Raabe’s Test

Abstract

SummaryAmong the techniques for determining the convergence of a series, Raabe’s test remains relatively unfamiliar to most mathematicians. We present several results relating to Raabe’s test that do not seem to be widely known, making the case that Raabe’s test should be featured more prominently in undergraduate calculus and analysis courses. In particular, we demonstrate that Raabe’s test may be viewed as an implicit comparison with a p-series, in the same manner that the ratio test and the root test constitute an implicit comparison with a geometric series. Moreover, Raabe’s test can sometimes simplify the process for determining conditional convergence.