Module 21  Sequences and Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 21.2: Series and Sequences of Partial Sums  
This lesson explores series and partial sums of infinite series. Series are used in many applications including integration, approximation, and the solution of differential equations. These applications arise in many disciplines, especially physics and chemistry. Defining a Series A series, which is not a list of terms like a sequence, is the sum of the terms in a sequence. If the series has a finite number of terms, it is a simple matter to find the sum of the series by adding the terms. However, when the series has an infinite number of terms the summation is more complicated and the series may or may not have a sum. Defining Partial Sums When working with series, it is often helpful to examine the partial sums that represent the sum of the first few terms. Suppose an infinite sequence is defined by . The terms of the sequence are . The first four partial sums of the associated infinite series are shown below, where s_{k} represents the sum of the first k terms of the sequence.
Each of the results shown above is a partial sum of the series which is associated with the sequence . Defining the Sequence of Partial Sums of a Series The partial sums of a series form a new sequence, which is denoted as {s_{1}, s_{2}, s_{3}, s_{4},...}. For the series given above, the sequence of partial sums is . If the sequence of partial sums for an infinite series converges to a limit L, then the sum of the series is said to be L and the series is convergent. Otherwise, the infinite series diverges. Finding Partial Sums of Series The TI89 summation function and the Sequence Graphing mode are useful tools in understanding the sequence of partial sums and convergence of a series. Using the Summation Function Partial sums can be computed with the sum function and may be used to help explore whether or not the infinite series converges. If it is convergent, the partial sums can also help estimate the sum of the series.
"4: ( sum" is the fourth item in this menu.
Beginning with Zero
Explore the partial sums of the series
. The first term in this series corresponds to
This result is called the fifth partial sum because the first five terms corresponding to k = 0, 1, 2, 3, 4 were added.
21.2.1 Approximate the tenth partial sum of the infinite series
. Graphing the Sequence of Partial Sums
The graph of the sequence of partial sums for the infinite series
can be created by defining the sequence of partial sums in the Y= Editor in Sequence Graph mode. You will need to enter the summation function from the catalog by pressing
, scrolling down the list to
Recall that n is the independent variable in Sequence Graphing mode.
Each point of this graph represents a partial sum.
This should be the same value for the tenth partial sum that you computed earlier on the Home Screen.
21.2.2 The graph of the sequence of partial sums levels off. What does this imply? Creating a Table of Values for the Partial Sums Make a table to show the partial sums.
Using the "Ask" Mode You can compute partial sums of your choosing by returning to the Table Setup dialog box and selecting the "ASK" mode.
The table provides further evidence that the series converges.
Using the Ratio Test The ratio test can confirm the convergence of this series. Theorem (Ratio Test) If for a series of positive terms, then the infinite series converges.
21.2.3 Define a function
and evaluate
.
21.2.4 The ratio test showed that the series converges. To what value do you think the series converges? 

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