taylor's series
LowAcademic / Technical (Mathematics, Physics, Engineering)
Definition
Meaning
A representation of a function as an infinite sum of terms, calculated from the values of its derivatives at a single point.
A fundamental concept in calculus for approximating complex functions with polynomials, enabling analysis, computation, and problem-solving in mathematics, physics, and engineering.
Linguistics
Semantic Notes
Specifically a mathematical term with no non-technical meaning. It is a proper noun, derived from the mathematician Brook Taylor, and is always used with the possessive form (Taylor's).
Dialectal Variation
British vs American Usage
Differences
No significant linguistic difference. Both dialects use the term identically in technical contexts. Spelling remains 'series' in both.
Connotations
Purely technical and neutral in both regions.
Frequency
Equally rare in general discourse but standard in university-level mathematics, physics, and engineering courses worldwide.
Vocabulary
Collocations
Grammar
Valency Patterns
The Taylor's series for [FUNCTION] about/around [POINT] is...We can approximate [FUNCTION] using a Taylor's series.Expanding [FUNCTION] in a Taylor's series yields...The first three terms of the Taylor's series are...Vocabulary
Synonyms
Neutral
Weak
Usage
Context Usage
Business
Virtually never used.
Academic
Core terminology in undergraduate mathematics, physics, and engineering courses. Used in textbooks, lectures, and research papers.
Everyday
Not used.
Technical
Essential for numerical analysis, modelling, control theory, and any field requiring function approximation or local analysis.
Examples
By Part of Speech
adjective
British English
- The Taylor's series coefficients were computed.
- We derived a Taylor's series solution.
American English
- The Taylor series coefficients were calculated.
- We found a Taylor series representation.
Examples
By CEFR Level
- In calculus, a Taylor's series can represent functions like sine or cosine as polynomials.
- The engineer used a Taylor's series to get an approximate solution to the complex equation.
- By expanding the potential energy function in a Taylor's series about the equilibrium point, we can analyse the system's stability.
- The proof relied on truncating the Taylor's series after the quadratic term and bounding the remainder.
Learning
Memory Aids
Mnemonic
Think of TAYLORing a suit: a Taylor's series TAILORS a complex function into a simpler, polynomial fit around a specific point.
Conceptual Metaphor
A FUNCTION IS A SUM OF BUILDING BLOCKS (the polynomial terms). UNDERSTANDING SOMETHING COMPLEX BY EXAMINING ITS IMMEDIATE SURROUNDINGS (the point of expansion).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- May be directly transliterated as 'ряд Тейлора' (ryad Teylora). The possessive 's' is lost in translation, which is fine. The key trap is confusing it with a generic 'series' (серия, ряд).
- Ensure the mathematical concept is distinguished from 'Fourier series' (ряд Фурье), which is a different type of expansion.
Common Mistakes
- Writing 'Taylor series' without the possessive apostrophe (though this is common and often accepted informally).
- Confusing it with a 'Maclaurin series' (which is a specific Taylor's series about zero).
- Using it to refer to any infinite series, rather than one specifically derived from derivatives at a point.
Practice
Quiz
What is a Taylor's series primarily used for?
FAQ
Frequently Asked Questions
Both are used, but 'Taylor's series' (possessive) is the original and more formally correct, named after Brook Taylor. 'Taylor series' is a common simplified variant.
A Maclaurin series is a special case of a Taylor series where the expansion is taken around the point zero (a=0).
No. A function must be infinitely differentiable at the expansion point to have a Taylor's series, and even then, the series may not converge to the function itself everywhere.
It is used extensively in physics for simplifying equations in small-oscillation problems, in engineering for control system linearisation, in computer science for numerical algorithms, and in economics for modelling behaviour near an equilibrium.