Alternating Series and Absolute Convergence
In this lesson, we learn about alternating series and absolute convergence. An alternating series is a series whose terms alternate in sign, and we can determine if it converges using the convergence theorem for alternating series. We can also estimate the remainder of a convergent alternating series using certain conditions. We explore the concepts of absolute and conditional convergence and see how they relate to alternating p series and geometric series. Finally, we learn about the ratio and root tests, which allow us to determine if the sum of a sub k converges absolutely or diverges.
Convergence theorem for alternating series. Estimation of the remainder. Absolute versus conditional convergence.
Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at http://online.math.uh.edu/HoustonACT/videocalculus/index.html.
- What is an alternating series?
- What is absolute convergence?
- What is the alternating harmonic series?
- How do you know if an alternating series will converge?
- Does the sum of (-1)^k/2k+1 converge or diverge?
- How do you estimate the nth remainder of a converging alternating series?
- How do you estimate the difference between the sum from k=1 to infinity of (-1)^(k-1)/k and its 100th partial sum?
- What is conditional convergence?
- What are alternating p-series?
- When do geometric series converge absolutely?
- For what values of x does the sum of x^k/k converge absolutely, converge conditionally, and diverge?
- Why does it matter if a series converges absolutely?
-
Staff Review
Alternating series and absolute convergence are defined and explained with several examples. Some very interesting and helpful examples are included. The geometric series, alternating p-series, ratio test, and root test are used in finding absolute and conditional convergence. This is a very useful lecture in Calculus.
