# Ch 22: Taylor Series

### About This Chapter

## 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. |

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

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### Other Chapters

Other chapters within the AP Calculus BC: Exam Prep course

- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Function Continuity
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Applying Integration Techniques
- Approximation of Definite Integrals
- Understanding Sequences & Series
- Series of Constants
- Using a Scientific Calculator for Calculus
- AP Calculus BC Flashcards