AP Calculus BC Review: Absolute and Conditional Convergence

Any given series either converges or diverges. But did you realize that there were different kinds of convergence? In this review article, we’ll take a look at the difference between absolute and conditional convergence. Along the way, we’ll see a few examples and discuss important special cases.Absolute and Conditional ConvergenceIf a series has a finite sum, then the series converges. Otherwise, the series diverges.However, we do make a distinction between series that converge very strongly, and those that only just barely converge. It all boils down to the signs of the terms.A series Σan converges absolutely if the series of the absolute values of its terms converges. That is, if Σ|an| also converges.A series Σan converges conditionally if the series converges but the associated series of absolute values Σ|an| diverges.Classifying SeriesIt’s important to realize that both absolute and conditional convergence are still types of convergence. It’s not that a conditionally convergent se...