Calculus BC – 10.13 Radius and Interval of Convergence of Power Series

In this lesson, Mr. Bean introduces the concepts of radius and interval of convergence for Power Series. He explains that these concepts are related but distinct, with the interval of convergence being a set of values where the series converges. The radius is the distance from the center of the series (often x=0) to the edge of the interval. Mr. Bean emphasizes the importance of identifying the center of the series as it is always part of the domain of convergence. He outlines three scenarios for convergence: to an interval, to all real numbers, or only to the center. The lesson also covers the use of the ratio test to determine the interval of convergence, with different outcomes based on whether the limit of the ratio test is less than one, equals zero, or equals infinity.

This section delves into the practical application of the ratio test to determine the interval of convergence for a given power series. The process involves simplifying the ratio of the (n+1)th term to the nth term and analyzing the limit as n approaches infinity. Depending on whether this limit is less than one, equals zero, or equals infinity, the series converges to an interval, all real numbers, or only to the center, respectively. Mr. Bean demonstrates this process with an example, showing how to simplify the expression and solve the resulting inequality to find the interval of convergence. He also stresses the importance of checking the endpoints to determine if they are included in the interval of convergence.

Continuing the practical application, Mr. Bean explains the necessity of checking the endpoints of the interval obtained from the ratio test. This step confirms whether the series converges at the endpoints or if it only converges within the interval. The explanation includes a detailed example where the series is evaluated at specific x-values to determine divergence or convergence. The process involves substituting the endpoints into the series and observing the behavior. If the series diverges at an endpoint, that endpoint is not included in the interval of convergence. This step is crucial for accurately determining the interval and the radius of convergence.

In this part, Mr. Bean addresses the challenge of simplifying more complex expressions in the ratio test. He provides a step-by-step walkthrough of simplifying a given power series term, emphasizing the importance of careful cancellation and factorization. After simplification, the focus shifts to determining the limit as n approaches infinity and solving the resulting inequality to find the interval of convergence. The example demonstrates how to handle a series with factorial terms and powers of x, leading to the identification of the series' convergence properties.

The final part of the script covers special cases in the convergence of power series. Mr. Bean explains what happens when the limit of the ratio test equals infinity, indicating that the series converges only at the center, with the radius being zero. Conversely, if the limit equals zero, the series converges for all values of x, implying an infinite radius of convergence. These scenarios are distinct from the typical interval convergence and require a different approach in analysis. The lesson concludes with a reminder of the importance of practice in mastering the application of the ratio test and simplifying complex expressions.