Series of Constants 

? 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. 

Problem Statement 

Determine if the series converges or diverges. If it converges, determine its value. 

 

Solution 

 

Explore the Sequence of Partial Sums 

 

Step

Result

Generate a sequence of the first few terms in the series.    To a copy of the general term of the series, append the sequence operator ($) as shown.  Press [Enter].    Convert this sequence to a list.    Right-click on the sequence and select Conversions>ToList    Obtain partial sums of the series.    Right-click on the list of terms and select Partial Sums.    Express the partial sums in floating-point form.    Right-click on the list of partial sums and select Approximate>5.    Plot the sequence of partial sums as a bar chart.    Right-click on the list of partial sums and select Statistics>Visualization>Bar Chart.  Select Quit to embed the graph in the document.

(1)         (2)         (3)         (4)

 

 

Formal Test for Convergence 

 

Step	Result
Use the Ratio Test to test the series for convergence.    Tasks>Browse: Calculus>Ratio Test. Reference the general term of the series by its equation label.	Ratio Test for , where is a sequence of positive constants:  General Term   >   (5)   Test Number   >   (6)   >

 

Since the Ratio Test yields a positive value that is less than 1, the series is convergent. 

 

If the Ratio Test fails (test number is 1), the Integral Test may be used. This test states that if for all and 

 

 

 

where represents the general term of the series and is treated as a continuous variable, then the series converges. 

 

Step	Result
Determine the convergence of the series using the integral test.     Use the integral template from the Expression palette to construct the integral. Here is treated as a continuous variable.	(7)

 

Since the value of the integral is finite, the series converges.  

 

Sum the Series 

 

Step	Result
Determine the value of the series.    From its position in the statement of the problem, Control-drag the series to a blank document block and press [Enter].	(8)

 

To obtain the value of the series from "first principles" recognize that the series is a linear combination of two geometric series, that is  

, 

and note that the sum of the geometric series is . 

 

Step	Result
Determine the value of the series from "first principles."    Note that the series is can be written as ,	(9)

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