The Taylor series is the power series:

\qquad \qquad f(x) = f(a) + \displaystyle \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...

The following applet can be used to graph the Taylor polynomial of degree n, Tn(x), and/or the remainder function, Rn(x), for the given function y=f(x), centered at x=a. Simply enter the function f(x) and the values a and 0 ≤ n ≤ 50. The applet can also be used to approximate the value of f(b) with Tn(b), for b near a. The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change this value by using the up/down arrow keys, clicking on the graph of y=f(x) or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

\qquad \qquad f(x) = f(a) + \displaystyle \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...

The following applet can be used to graph the Taylor polynomial of degree n, Tn(x), and/or the remainder function, Rn(x), for the given function y=f(x), centered at x=a. Simply enter the function f(x) and the values a and 0 ≤ n ≤ 50. The applet can also be used to approximate the value of f(b) with Tn(b), for b near a. The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change this value by using the up/down arrow keys, clicking on the graph of y=f(x) or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

## Software/Applets used on this page

This page uses an applet from David Little (Lecturer in the Mathematics Department, Penn State University, USA) and is used with his permission.

## Glossary

### degree

On a graph: the number of edges incident to a vertex.

In geometry: a measure of angle

In geometry: a measure of angle

### function

A rule that connects one value in one set with one and only one value in another set.

### graph

A diagram showing a relationship between two variables.

The diagram shows a vertical y axis and a horizontal x axis.

The diagram shows a vertical y axis and a horizontal x axis.

### polynomial

an expression which has whole-numbered powers of the variable, x, multiplied by coefficients, and added together.

Examples: y = 2x, y = 1+x²,y = 3-2x+7x³

Examples: y = 2x, y = 1+x²,

### power series

a series whose terms form a power sequence, often obtained from Maclaurin or Taylor series

### remainder

The number remaining after one polynomial is divided into another an exact number of times.

### series

the sum of terms in a sequence

### taylor series

a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AP Calculus BC (USA) | 5 | Maclaurin and Taylor Series | Taylor series | - |

AQA A2 Further Maths 2017 | Pure Maths | Extra | Taylor Series | - |

AQA AS/A2 Further Maths 2017 | Pure Maths | Extra | Taylor Series | - |

Edexcel A-Level (UK - Pre-2017) | FP2 | Maclaurin and Taylor Series | Taylor series | - |

Edexcel A2 Further Maths 2017 | Further Pure 1 | Methods in Calculus | Taylor Series | - |

Edexcel AS/A2 Further Maths 2017 | Further Pure 1 | Methods in Calculus | Taylor Series | - |

I.B. Higher Level | 9 | Maclaurin and Taylor Series | Taylor series | - |

Methods (UK) | M10 | Maclaurin and Taylor Series | Taylor series | - |

OCR A2 Further Maths 2017 | Pure Core | Extra | Taylor Series | - |

OCR MEI AS Further Maths 2017 | Numerical Methods | Extra | Taylor Series | - |

Scottish Advanced Highers | M3 | Maclaurin and Taylor Series | Taylor series | - |

Scottish (Highers + Advanced) | AM3 | Maclaurin and Taylor Series | Taylor series | - |

Universal (all site questions) | M | Maclaurin and Taylor Series | Taylor series | - |

WJEC A-Level (Wales) | FP3 | Maclaurin and Taylor Series | Taylor series | - |