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Course: Calculus III
Topic: Power Series
Subtopic: Representing Functions as Power Series

Overview

Taylor polynomials can provide accurate approximations to many functions. The approximation is improved by increasing the polynomial's degree. If we let the degree increase to form an infinite polynomial we produce power series. In this section we represent a variety of functions as power series, learn to combine power series, and discuss when a power series is convergent.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 12.2.1 Write a geometric series as a power series
• 12.2.2 Write a variety of basic functions as power series centered at x=0
• 12.2.3 Understand the convergence of a power series
• 12.2.4 Find the interval of convergence and radius of convergence of a give power series
• 12.2.5 Know the three ways in which a power series converges
• 12.2.6 Combine power series (e.g. find their sum, difference, or composition) and perform multiplication by a power
• 12.2.7 Know that a power series can be differentiated or integrated term-by-term

Terminology

Define: Each of the following in terms of a power series - coefficients, center, convergence, interval of convergence, radius of convergence

Supplemental Resources (recommended)

View prezi covering Power Series and Convergence from Michael Maestas

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Taylor Series

Selwyn Hollis's Video Calculus: Power Series
(Skip through the ratio and root tests included in this video.)

Patrick JMT Just Math Tutorials:
Power Series: Differentiating and Integrating
Power Series: Multiplying and Dividing
Taylor and Maclaurin Series – Example 1

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down middle column to "Infinite Series" for related titles.