Chapter 11: Sequences, Series, and Power Series

Students learn how to determine whether a sequence approaches a finite value, oscillates, or grows without bound. Infinite series are then introduced as sums of sequence terms, starting with geometric series and their convergence criteria. The chapter covers the n-th term test for divergence, the integral test, the comparison and limit comparison tests, the alternating series test, and the ratio and root tests for absolute convergence. Special attention is given to the concept of conditional convergence and how rearranging terms can affect the sum of a series. Power series are introduced as functions represented by infinite sums, with discussions of interval and radius of convergence and how to determine them. Taylor and Maclaurin series are presented as powerful tools for approximating functions, along with methods for finding their expansions and error bounds using Taylor’s theorem. Applications include function approximation, solving differential equations, and evaluating limits and integrals that are otherwise intractable. By the end of the chapter, students can analyze, test, and apply infinite series to model complex mathematical and physical phenomena.