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2: Power Series

Last updated

Apr 1, 2025
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- 1.7: Chapter 1 Review Exercises
- 2.1: Power Series and Functions

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A power series (in one variable) is an infinite series. Any polynomial can be easily expressed as a power series around any center c, although most of the coefficients will be zero since a power series has infinitely many terms by definition. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials. The content in this Textmap's chapter is complemented by Guichard's Calculus Textmap.

When winning a lottery, sometimes an individual has an option of receiving winnings in one lump-sum payment or receiving smaller payments over fixed time intervals. For example, you might have the option of receiving 20 million dollars today or receiving 1.5 million dollars each year for the next 20 years. Which is the better deal? Certainly 1.5 million dollars over 20 years is equivalent to 30 million dollars. However, receiving the 20 million dollars today would allow you to invest the money.

A photograph shows a flat surface covered with $100 bills. — Figure $\PageIndex{}$ : An image showing multiple U.S. $$100$ bills arranged in a grid-like pattern. The bills feature the portrait of Benjamin Franklin and other recognizable details, such as the green treasury seal and serial numbers (Credit:CC-BY-NC-SA,work by Ervins Strauhmanis's,www.flickr.com)

Alternatively, what if you were guaranteed to receive 1 million dollars every year indefinitely (extending to your heirs) or receive 20 million dollars today. Which would be the better deal? To answer these questions, you need to know how to use infinite series to calculate the value of periodic payments over time in terms of today’s dollars.

An infinite series of the form

$\sum_{n=0}^∞c_nx^n \nonumber$

is known as a power series. Since the terms contain the variable $x$ , power series can be used to define functions. They can be used to represent given functions, but they are also important because they allow us to write functions that cannot be expressed any other way than as “infinite polynomials.” In addition, power series can be easily differentiated and integrated, thus being useful in solving differential equations and integrating complicated functions. An infinite series can also be truncated, resulting in a finite polynomial that we can use to approximate functional values. Power series have applications in a variety of fields, including physics, chemistry, biology, and economics. As we will see in this chapter, representing functions using power series allows us to solve mathematical problems that cannot be solved with other techniques.

Thumbnail: The graph shows the function $\displaystyle y=sinx$ and the Maclaurin polynomials $\displaystyle p_1,p_3$ and $\displaystyle p_5$ . (CC BY--NC-SA 4.0; OpenStax).

Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

- 2.1: Power Series and Functions
  A power series is a type of series with terms involving a variable. More specifically, if the variable is x, then all the terms of the series involve powers of x. As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions. In this section we define power series and show how to determine when a power series converges and when it diverges. We also show how to represent certain functions using power
  2.1E: Exercises for Section 2.1
- 2.2: Properties of Power Series
  Power series can be combined, differentiated, or integrated to create new power series. This capability is particularly useful for a couple of reasons. First, it allows us to find power series representations for certain elementary functions, by writing those functions in terms of functions with known power series. Second, it allows us to define new functions that cannot be written in terms of elementary functions. This capability is particularly useful for solving differential equations.
  2.2E: Exercises for Section 2.2
- 2.3: Taylor and Maclaurin Series
  Here we discuss power series representations for other types of functions. In particular, we address the following questions: Which functions can be represented by power series and how do we find such representations? If we can find a power series representation for a particular function ff and the series converges on some interval, how do we prove that the series actually converges to f?
  2.3E: Exercises for Section 2.3
- 2.4: Working with Taylor Series
  In this section we show how to use those Taylor series to derive Taylor series for other functions. We then present two common applications of power series. First, we show how power series can be used to solve differential equations. Second, we show how power series can be used to evaluate integrals when the antiderivative of the integrand cannot be expressed in terms of elementary functions.
  2.4E: Exercises for Section 2.4
- 2.5: Chapter 2 Review Exercises

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