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: Taylor Polynomials and Approximations

Overview

A Power Series is basically an infinite polynomial of the form f(x) = a_0(x-c)^0+ a_1(x-c)^1+ a_2(x-c)^2+ ... where each term depends on variable x written in ascending power order. Taylor Series provide a formula for writing the power series for a given function. The result is a Taylor Polynomial. These polynomials can be used to approximate non-polynomial functions such as radical, exponential, or trigonometric. Polynomial functions are often easier to work with (e.g. differentiate) than transcendental functions so using polynomials to approximate them can be helpful in calculus and analysis.

Objectives

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

• 12.1.1 Know what a power series is and what it does for a non-polynomial function
• 12.1.2 Perform linear and quadratic approximations using power series
• 12.1.3 Build longer and longer polynomials that can be used to approximate functions with increased accuracy near a specific x-value
• 12.1.4 Know the definition of an "n-th order Taylor Polynomial for f(x) with center at a" and be able to build the polynomial's coefficients
• 12.1.5 Perform approximations with Taylor Polynomials of given order
• 12.1.6 Write Taylor Polynomials in both expanded form and using summation notation
• 12.1.7 Estimate a Real number using a Taylor Polynomial
• 12.1.8 Use formulae to find the remainder in a Taylor Polynomial and estimate the remainder for a particular function
• 12.1.9 Understand and be able to apply Taylor's Theorem

Terminology

Terms you should be able to define: power series, Taylor Series, Taylor Polynomial, coefficients of Taylor Polynomial,remainder, error, magnitude of error, absolute error, maximum error

Formulae to have in your notes: Taylor Series, remainder in a Taylor Polynomial, estimate of the remainder theorem, Taylor's Theorem

Supplemental Resources (recommended)

Watch Essence of Calculus - Taylor Series video from 3Blue1Brown as intuitive intro to Taylor Series

View slides overviewing Power Series and Taylors Theorem from Dr. Thomases, UC Davis

Supplemental Resources (optional)

View slides on Power Series and Taylor Series (inc. "remainder" info) from Dr. DeTurck, U Penn

Paul's OL Notes - Calc II: Power Series and Power Series with Functions

Selwyn Hollis's Video Calculus: Taylor's Theorem

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.