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.
Software/Applets used on this page
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.
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
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 | - |