Course: Calculus III
Topic: Power Series
Subtopic: Taylor Polynomials and Approximations


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.


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


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)

Download/Print Prof. Keely's Frequently Used Power Series Sheet

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

Patrick JMT Just Math Tutorials:
Power Series: Finding the Interval of Convergence
Radius of Convergence for a Power Series
Power Series Representation of a Function

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.