Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus III
Topic: Power Series
Subtopic: Calculus of Power Series

Overview

Now that we are able to write a variety of functions as power series, we can use the power series form of these functions to perform calculus processes such as finding limits, differentiating, and integrating. Doing calculus on the power series is often much easier than on the original function.

Objectives

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

• 12.4.1 Evaluate a limit by substituting the power series in for the function and finding the limit of it
• 12.4.2 Differentiate a function by substituting the power series in for the function and differentiating it
• 12.4.3 Solve a basic differential equation by substituting a power series in for the function and its derivative(s) and solving the resulting equation
• 12.4.4 Integrate a function by substituting the power series in for the function and integrating it
• 12.4.5 Approximate the value of a definite integral (to within a specific error) by substituting the power series on for the function and integrating it
• 12.4.6 Represent Real numbers using power series (eg. substitute x-values into a convergent power series to represent a familiar Real number such as pi)
• 12.4.7 Represent functions as power series (eg. given a power series identify the function it represents)

Supplemental Resources (recommended)

A worthy (and very cool!) expansion of this material, one with current applications in a variety of STEAM fields (science, technology, engineering, arts, mathematics) is Fourier Series. Lots of good reads on the web. I recommend that you at least read An Introduction to Fourier Series by Dale Hoffman (Bellevue College in WA) and Fourier Series Introduction from Interactive Mathematics.

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Application of Series

Patrick JMT Just Math Tutorials: Using Series to Evaluate Limits

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.