Taylor series | GPTKB 2.0

All labels observed (6)

Label	Occurrences
Taylor series canonical	8
Maclaurin series	2
La série de Taylor et son prolongement analytique	1
Taylor series expansion	1
Taylor's theorem	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	Maclaurin series ⓘ linked to: Taylor 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

Referenced by (14)

Full triples — surface form annotated when it differs from this entity's canonical label.

generalized binomial theorem → relatedConcept → Taylor series ⓘ

Jacques Hadamard → notableWork → La série de Taylor et son prolongement analytique ⓘ

linked to: Taylor series

Taylor series → specialCase → Maclaurin series ⓘ

linked to: Taylor series

Colin Maclaurin → notableWork → Maclaurin series ⓘ

linked to: Taylor series

Brook Taylor → knownFor → Taylor series ⓘ

Brook Taylor → knownFor → Taylor's theorem ⓘ

linked to: Taylor series

Essai sur l’étude des fonctions données par leur développement de Taylor → mainSubject → Taylor series ⓘ

Itô–Taylor expansion → generalizes → Taylor series ⓘ

Volterra series → generalizes → Taylor series ⓘ

Cauchy’s mean value theorem → relatedTo → Taylor’s theorem ⓘ

linked to: Taylor series

Lax–Wendroff method → usesMathematicalTool → Taylor series expansion ⓘ

linked to: Taylor series

Methodus Incrementorum Directa et Inversa → introducedConcept → Taylor series ⓘ

Brook Taylor → knownFor → Taylor series ⓘ

subject linked to: Linear Perspective

Brook Taylor → notableWork → Taylor series ⓘ

subject linked to: John Taylor

All labels observed (6)

How this entity was disambiguated Show

Statements (50)

How these facts were elicited Show

Referenced by (14)

How this entity was disambiguated

How these facts were elicited