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Ch 22: Taylor Series

About This Chapter

Watch these video lessons to understand Taylor series and the various types of functions which can be evaluated using them. Complete the self-assessments to gauge retention of key topics.

Taylor Series - Chapter Summary and Learning Objectives

The lessons in this chapter explore how a function's derivatives can be used to express its infinite sums. Taylor series are a versatile tool for evaluating and computing series, so every lesson here deals with a certain aspect of Taylor series and its offshoots. Complete the video lessons and assessments to boost your understanding of series calculations in calculus. In this chapter our instructors discuss:

  • The definition and formula for Taylor series
  • Coefficients and polynomials of Taylor series
  • Evaluating functions of a complex variable and polynomial approximation with Taylor series
  • The definition and formula for Maclaurin series
  • Interval and radius of convergence with Power series
  • Assessing Taylor polynomials using the Lagrange error bound

Video Objective
Taylor Series: Definition, Formula & Examples In this lesson our instructors outline how Taylor series represent functions using the derivatives of a function at a point.
Taylor Series, Coefficients & Polynomials: Definition, Equations & Examples The Taylor series is further evaluated in this video on coefficients and polynomials within Taylor series.
Taylor Series for Functions of a Complex Variable More discussion and application of Taylor series in this brief video.
Taylor Polynomial Approximation Instructors demonstrate methods for finding approximations of polynomial values by graphing Taylor series.
Maclaurin Series: Definition, Formula & Examples Here you will learn about how the Maclaurin series is a special case of the Taylor series and identify the related formula.
Manipulations & Shortcuts for Taylor Series Computations Watch this video to learn some handy tricks to solve complex problems through substitution, differentiation, and other methods.
Functions Defined by Power Series In this brief lesson on power series you will expand from looking at series with numbers only into working with series containing a variable.
Interval & Radius of Convergence of Power Series Learn to assess different types of convergence, which are dependent on the values of x in power series, with this video lesson.
Lagrange Error Bound for Taylor Polynomials This video describes how to find and evaluate Taylor polynomials using the Lagrange error bound.

7 Lessons in Chapter 22: Taylor Series
Test your knowledge with a 30-question chapter practice test
Taylor Series: Definition, Formula & Examples

1. Taylor Series: Definition, Formula & Examples

In mathematics, we sometimes gain insights by expressing functions in different ways. One of those ways is the Taylor series. In this lesson, we will define the Taylor series and use examples to demonstrate its use.

Taylor Series, Coefficients & Polynomials: Definition, Equations & Examples

2. Taylor Series, Coefficients & Polynomials: Definition, Equations & Examples

A Taylor series is an infinite series representing a function. In this lesson we explore how to use a finite number of terms of a Taylor series to approximate an integration result.

Taylor Series for Functions of a Complex Variable

3. Taylor Series for Functions of a Complex Variable

This lesson explores how analytic functions can be expressed as Taylor series. We use the concepts of complex differentiable functions and Cauchy-Riemann equations.

Maclaurin Series: Definition, Formula & Examples

4. Maclaurin Series: Definition, Formula & Examples

In this video lesson, you will learn how the Maclaurin series is a special case of the Taylor series. You'll also discover what some common Maclaurin series are for functions such as e^x and sin x.

Power Series: Formula & Examples

5. Power Series: Formula & Examples

A power series gives us what we call an infinite polynomial on our variable x and can be used to define a wide variety of functions. Learn the definition of power series as well as several examples of functions that can be described by them.

Power Series in X & the Interval of Convergence

6. Power Series in X & the Interval of Convergence

A power series is often used in physics and engineering to approximate functions. In this lesson, we'll explore the power series in x and show how to find the interval of convergence.

Functions Defined by Power Series

7. Functions Defined by Power Series

Do you remember working with polynomials? Imagine a polynomial that is infinitely long. Instead of having a finite degree, it has an infinite degree. Such an infinite polynomial is called a power series. In this lesson, we examine traits of power series.

Chapter Practice Exam
Test your knowledge of this chapter with a 30 question practice chapter exam.
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Practice Final Exam
Test your knowledge of the entire course with a 50 question practice final exam.
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