# 14.2.7: Power Series

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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.

• 14.2.7.1: Prelude to Power Series
Power series can be used to define functions and they allow us to write functions that cannot be expressed any other way than as “infinite polynomials.” An infinite series can also be truncated, resulting in a finite polynomial that we can use to approximate functional values. Representing functions using power series allows us to solve mathematical problems that cannot be solved with other techniques.
• 14.2.7.2: 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
• 14.2.7.3: 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?
• 14.2.7.4: 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.
• 14.2.7.5: Fourier Series and Transform
The Fourier transform is the underlying principle for frequency-domain description of signals: it allows a time-domain signal to be transformed into a (complex) frequency-domain version, and to be transformed back as necessary.
• 14.2.7.6: The Laplace Transform
A basic overview of the role of the Laplace transform in analyzing dynamic systems, the Convolution Theorem, and in solving differential equations.