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This lecture from M.I.T. is nearly 50 minutes long delivered by Professor David Jerison and covers the application of Taylor's series to a problem about stacking bricks. It is aimed at undergraduate students.

These lectures will give you some idea of how this subject is treated at university level.

## Summary/Background

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.

Free lecture notes, exams, and videos are available from Massachusetts Institute of Technology at MIT. No registration required.

## 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

### 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

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus BC (USA)5Maclaurin and Taylor SeriesTaylor series-
AQA A2 Further Maths 2017Pure MathsExtraTaylor Series-
AQA AS/A2 Further Maths 2017Pure MathsExtraTaylor Series-
Edexcel A-Level (UK - Pre-2017)FP2Maclaurin and Taylor SeriesTaylor series-
Edexcel A2 Further Maths 2017Further Pure 1Methods in CalculusTaylor Series-
Edexcel AS/A2 Further Maths 2017Further Pure 1Methods in CalculusTaylor Series-
I.B. Higher Level9Maclaurin and Taylor SeriesTaylor series-
Methods (UK)M10Maclaurin and Taylor SeriesTaylor series-
OCR A2 Further Maths 2017Pure CoreExtraTaylor Series-
OCR MEI AS Further Maths 2017Numerical MethodsExtraTaylor Series-
Scottish Advanced HighersM3Maclaurin and Taylor SeriesTaylor series-
Scottish (Highers + Advanced)AM3Maclaurin and Taylor SeriesTaylor series-
Universal (all site questions)MMaclaurin and Taylor SeriesTaylor series-
WJEC A-Level (Wales)FP3Maclaurin and Taylor SeriesTaylor series-