Taylor series
E146426
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Taylor series canonical | 3 |
| Maclaurin series | 2 |
| La série de Taylor et son prolongement analytique | 1 |
| Taylor's theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1286004 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Taylor series Context triple: [generalized binomial theorem, relatedConcept, Taylor series]
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
C.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
D.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
-
E.
generalized binomial theorem
The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Taylor series Target entity description: A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
C.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
D.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
-
E.
generalized binomial theorem
The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Taylor series
ⓘ
mathematical concept ⓘ power series expansion ⓘ |
| appliesTo |
complex-valued functions
ⓘ
multivariable functions ⓘ real-valued functions ⓘ vector-valued functions ⓘ |
| expandsAround |
0
ⓘ
a single point, often denoted a ⓘ |
| field |
calculus
ⓘ
complex analysis ⓘ mathematical analysis ⓘ real analysis ⓘ |
| generalization | multivariable Taylor series ⓘ |
| hasComponent |
coefficients given by derivatives divided by factorials
ⓘ
powers of the variable measured from an expansion point ⓘ |
| hasDefinition | representation of a function as an infinite sum of terms derived from its derivatives at a single point ⓘ |
| hasExample |
expansion of cos x as sum from n=0 to infinity of (-1)^n x^(2n) divided by (2n)!
ⓘ
expansion of e^x as sum from n=0 to infinity of x^n divided by n! ⓘ expansion of sin x as sum from n=0 to infinity of (-1)^n x^(2n+1) divided by (2n+1)! ⓘ |
| hasHistoricalNote | introduced by Brook Taylor in the early 18th century ⓘ |
| hasProperty |
can be truncated to give polynomial approximations
ⓘ
can represent analytic functions exactly within radius of convergence ⓘ coefficients uniquely determined by derivatives at the expansion point ⓘ may converge only on a certain interval around the expansion point ⓘ may fail to converge to the function outside its radius of convergence ⓘ truncation error can be bounded using remainder term ⓘ |
| namedAfter | Brook Taylor ⓘ |
| relatedTo |
Taylor polynomial
ⓘ
Taylor remainder ⓘ analytic function ⓘ asymptotic expansion ⓘ power series ⓘ radius of convergence ⓘ |
| specialCase |
Taylor series
self-linksurface differs
ⓘ
surface form:
Maclaurin series
|
| usedFor |
approximating functions near a point
ⓘ
approximating transcendental functions ⓘ error estimation via remainder terms ⓘ local analysis of functions ⓘ numerical computation ⓘ series solutions of ordinary differential equations ⓘ solving differential equations ⓘ |
| usedIn |
approximation theory
ⓘ
computer science ⓘ economics ⓘ engineering ⓘ numerical analysis ⓘ perturbation theory ⓘ physics ⓘ |
| uses | derivatives of a function at a point ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Taylor series Description of subject: A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
La série de Taylor et son prolongement analytique
this entity surface form:
Maclaurin series
this entity surface form:
Maclaurin series
this entity surface form:
Taylor's theorem