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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. You can get a better display of the maths by downloading special TeX fonts from jsMath. In the meantime, we will do the best we can with the fonts you have, but it may not be pretty and some equations may not be rendered correctly.

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:

SyllabusModuleSectionTopic
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