Module 22  Power Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 22.1: Power Series  
In this lesson you will study several power series and discover that on the intervals where they converge, they are equal to certain well known functions. Defining Power Series A power series is a series in which each term is a constant times a power of x or a power of (x  a). Suppose each c_{k} represents some constant. The infinite series
is a power series centered at x = 0. The infinite series
is a power series centered at x = a. Finding Partial Sums of a Power Series Consider the power series
Although you cannot enter infinitely many terms of this series in the Y= Editor, you can graph partial sums of the series because each partial sum is a polynomial with a finite number of terms. Graphing Partial Sums of a Power Series
22.1.1 Graph the second, third and fourth partial sums of the power series
in a [5, 5] x Defining an Infinite Geometric Series Recall that an infinite geometric series can be written as a + ar + ar^{2} + ar^{3} + ... + ar^{k} + ..., where a represents the first term and r represents the common ratio of the series. If  r  < 1, the infinite geometric series a + ar + ar^{2} + ar^{3} + ... + ar^{k} + ... converges to . The power series
is a geometric series with first term 1 and common ratio x. This means that the power series converges when  x  < 1 and converges to on the interval (1, 1). Visualizing Convergence The graphs of several partial sums can illustrate the interval of convergence for an infinite series.
On the interval (1,1) the partial sums are close to . The interval (1,1) is called the interval of convergence for this power series. As the number of terms in the partial sums increase, the partial sums converge to .
22.1.2 Graph the tenthdegree partial sum of y = 1/(1  x) in a [2, 2] x [2, 8] viewing window. Use the Investigating Another Power Series Consider the power series
22.1.3 Examine the graphs of the partial sums listed above on the interval 6 < x < 6. The partial sums seem to converge to what familiar function?
22.1.4 For the partial sums in Question 22.1.2 graph the familiar function along with the partial sums in a [10, 10] x [2, 2] window. Change the graphing style of the familiar function to "thick". Adding More Terms to Partial Sums As more terms are added to form successive partial sums, more turning points appear in each successive graph and the power series converges to the function y = sin x on the interval (– , ). Illustrating Convergence of In Lesson 22.2 you will see that the power series converges to e^{x}. Graphing the second, fifth, eighth and eleventhdegree partial sums of the power series illustrates that the infinite series converges to y = e^{x}.
22.1.5 What seems to happen to the interval where the partial sums are approximately equal to e^{x} as more terms are added in successive partial sums? 

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