Power Series 

? Maplesoft, a division of Waterloo Maple Inc., 2007 

Introduction 

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos. 

The steps in the document can be repeated to solve similar problems. 

Problem Statement 

Obtain the radius of convergence and the interval of convergence for the power series . 

Solution 

Radius of Convergence 

 

Step	Result
Compute the radius of convergence.    Enter the expression for the general coefficient and press [Enter].           Tools>Tasks>Browse>Calculus>Radius of Convergence. Click Insert Minimal  Content. Enter in the relevant information.	(1)       Ratio-Test Method for Radius of Convergence of and fixed integers, and positive:  General term   >   (2)   Enter , the coefficient of in the power of in the general term:  >   (3)   Radius of Convergence  >   (4)   >

 

Thus the radius of convergence of the series is .  

 

Interval of Convergence 

 

Note that the center of the interval of convergence is . To this, add and subtract , where is the radius of convergence. Thus, the interval of convergence is at least  

 

 

 

To determine the convergence of the series at the endpoints of this interval, substitute into the general term for the series, the endpoint values for ; then test the convergence of the resulting series. At , the series takes the form  

 

. 

 

At , the series takes the form  

 

. 

 

Step	Result
Test convergence at the point .     Use the Integral test to determine the convergence of the series. Use the definite integral template in the Expression palette to define the integral.	(5)
Test convergence at the point .     Use the alternating series test to test the convergences of the relevant series. Use the limit template from the Expression palette  to enter the expression for the limit. Press [Enter] to evaluate.        Show that the general term is (eventually) decreasing by considering it as a function of and taking its derivative. Enter the expression for the function, press [Enter], right-click, Differentiate>.Right-click, Simplify.	(6)         (7)     (8)     (9)

 

Since the Integral test indicates that at the series diverges.  

 

At , and when considered as a function of where is taken to be a continuous variable, the function is decreasing for all . Thus, the general term is decreasing for all . By the Alternating Series Test, the series is conditionally convergent at (but not absolutely convergent). Thus, the interval of convergence for the series is  

 

 

 

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