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f(x) = f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\frac{x^3}{3!}f^{(3)}(0)+...+\frac{x^r}{r!}f^{(r)}(0)+...+\frac{x^n}{n!}f^{(n)}(0)

## Summary/Background

The power series is \sin x = \displaystyle \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}.
The graphs shows approximations to \sin x for n = 0, 1, 2 and 3.
In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named in honour of English mathematician Brook Taylor. If the series uses the derivatives at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin (February 1698 – 14 June 1746).

Maclaurin was a Scottish mathematican who published the first systematic exposition of Newton's methods, written as a reply to Berkeley's attack on the calculus for its lack of rigorous foundations.

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## Glossary

### calculus

the study of change; a major branch of mathematics that includes the study of limits, derivatives, rates of change, gradient, integrals, area, summation, and infinite series. Historically, it has been referred to as "the calculus of infinitesimals", or "infinitesimal calculus".
There are widespread applications in science, economics, and engineering.

### function

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

### limit

the value that a function f(x) approaches as the variable x approaches a value such as 0 or infinity

### maclaurin series

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

### power series

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

### 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

### union

The union of two sets A and B is the set containing all the elements of A and B.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus BC (USA)5Maclaurin and Taylor SeriesMaclaurin series-
AQA A-Level (UK - Pre-2017)FP3Maclaurin and Taylor SeriesMaclaurin series-
AQA A2 Further Maths 2017Pure MathsFurther Sequences and SeriesFurther Maclaurin Series-
AQA AS Further Maths 2017Pure MathsSequences, Series and ProofMaclaurin Series-
AQA AS/A2 Further Maths 2017Pure MathsSequences, Series and ProofMaclaurin Series-
CCEA A-Level (NI)FP2Maclaurin and Taylor SeriesMaclaurin series-
Edexcel A-Level (UK - Pre-2017)FP2Maclaurin and Taylor SeriesMaclaurin series-
Edexcel A2 Further Maths 2017Core Pure MathsFurther SeriesMaclaurin Series-
Edexcel AS/A2 Further Maths 2017Core Pure MathsFurther SeriesMaclaurin Series-
I.B. Higher Level9Maclaurin and Taylor SeriesMaclaurin series-
Methods (UK)M10Maclaurin and Taylor SeriesMaclaurin series-
OCR A-Level (UK - Pre-2017)FP2Maclaurin and Taylor SeriesMaclaurin series-
OCR A2 Further Maths 2017Pure CoreCalculusMaclaurin Series-
OCR MEI A2 Further Maths 2017Core Pure BSeriesMaclaurin Series-
OCR-MEI A-Level (UK - Pre-2017)FP2Maclaurin and Taylor SeriesMaclaurin series-
Scottish Advanced HighersM3Maclaurin and Taylor SeriesMaclaurin series-
Scottish (Highers + Advanced)AM3Maclaurin and Taylor SeriesMaclaurin series-
Universal (all site questions)MMaclaurin and Taylor SeriesMaclaurin series-
WJEC A-Level (Wales)FP3Maclaurin and Taylor SeriesMaclaurin series-