14.2.7.1: Prelude to Power Series

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An infinite series of the form

\[\sum_{n=0}^∞c_nx^n\]

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.

Contributors and Attributions

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Modifications to compact this for review. Scott Johnson, Scott Sinex, and Joshua Halpern.