Taylor Series Ontology

FLOW ALGEBRA
TITLE: Taylor Series Ontology
VERSION: 4.0
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/taylor-series-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; source page content not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
DATE: 2026-10-13

1. PURPOSE
Represent Taylor series as typed mathematical entities, formal relationships, verifiable rules, and machine-consumable semantic structures. Distinguish a formal Taylor expansion from its convergence and from equality to the generating function.

2. ENTITY TYPES
ENTITY TaylorSeries <: PowerSeries
ENTITY TaylorPolynomial <: Polynomial
ENTITY ExpansionCentre <: Scalar
ENTITY Coefficient <: Scalar
ENTITY Derivative <: Function
ENTITY Remainder <: Function
ENTITY RadiusOfConvergence <: ExtendedNonnegativeReal
ENTITY IntervalOfConvergence <: SubsetOfReal
ENTITY DiskOfConvergence <: SubsetOfComplex
ENTITY AnalyticFunction <: Function
ENTITY SmoothFunction <: Function
ENTITY MaclaurinSeries <: TaylorSeries
ENTITY PartialSum <: Polynomial
ENTITY ConvergenceCriterion <: Predicate
ENTITY TruncationError <: NonnegativeReal
ENTITY Domain <: Set
ENTITY Variable <: Scalar
ENTITY Index <: NonnegativeInteger
ENTITY Factorial <: PositiveInteger

3. TYPED RELATIONSHIPS
TaylorSeries --IS_A--> PowerSeries
MaclaurinSeries --IS_A--> TaylorSeries
TaylorSeries --EXPANDS_AT--> ExpansionCentre
TaylorSeries --APPROXIMATES--> Function
TaylorSeries --HAS_COEFFICIENT--> Coefficient
TaylorSeries --HAS_PARTIAL_SUM--> TaylorPolynomial
TaylorPolynomial --HAS_REMAINDER--> Remainder
TaylorSeries --HAS_RADIUS--> RadiusOfConvergence
TaylorSeries --HAS_CONVERGENCE_DOMAIN--> IntervalOfConvergence | DiskOfConvergence
AnalyticFunction --EQUALS_LOCALLY--> TaylorSeries
SmoothFunction --DOES_NOT_NECESSARILY_EQUAL--> TaylorSeries

4. FORMAL DEFINITIONS
DEF TAYLOR_COEFFICIENT:
  GIVEN f with derivatives of all orders at a,
  c_n = f^(n)(a)/n!, n >= 0.
DEF TAYLOR_SERIES:
  T_a[f](x) = SUM(n=0..infinity) c_n (x-a)^n.
DEF TAYLOR_POLYNOMIAL:
  P_N(x) = SUM(n=0..N) f^(n)(a)(x-a)^n/n!.
DEF REMAINDER:
  R_N(x) = f(x)-P_N(x).
DEF MACLAURIN:
  T_0[f](x) = SUM(n=0..infinity) f^(n)(0)x^n/n!.
DEF SERIES_CONVERGENCE:
  T_a[f] converges at x iff its partial sums P_N(x) have a finite limit.
DEF REPRESENTATION:
  f(x) = T_a[f](x) iff lim(N->infinity) R_N(x)=0.
DEF ANALYTIC_AT_A:
  f is analytic at a iff there exists r>0 such that f(x)=T_a[f](x) for |x-a|<r (real case); complex holomorphic functions have local Taylor representations.

5. RADIUS OF CONVERGENCE
FORMULA CAUCHY_HADAMARD:
  1/R = limsup(n->infinity) |c_n|^(1/n), with 1/0=infinity and 1/infinity=0.
FORMULA RATIO:
  If lim(n->infinity) |c_n/c_(n+1)| = L exists for eventually nonzero coefficients, then R=L.
RULE INTERIOR_CONVERGENCE:
  |x-a|<R -> absolute convergence.
RULE EXTERIOR_DIVERGENCE:
  |x-a|>R -> divergence.
RULE BOUNDARY_INDEPENDENCE:
  |x-a|=R -> test each boundary point separately.
RULE COMPLEX_DISK:
  For complex z, |z-a|<R gives an open disk of absolute convergence.

6. TAYLOR'S THEOREM AND ERRORS
FORMULA LAGRANGE_REMAINDER:
  Under the standard continuity/differentiability hypotheses on the interval between a and x,
  R_N(x) = f^(N+1)(xi)(x-a)^(N+1)/(N+1)! for some xi between a and x.
FORMULA INTEGRAL_REMAINDER:
  If f^(N) is absolutely continuous on the interval and f^(N+1) integrable,
  R_N(x) = (1/N!) INTEGRAL(t=a..x) f^(N+1)(t)(x-t)^N dt.
BOUND:
  If |f^(N+1)(t)|<=M on the segment from a to x, then |R_N(x)|<=M|x-a|^(N+1)/(N+1)!.
CAUTION:
  Existence of derivatives of every order at a does not alone imply equality of f and its Taylor series.

7. OPERATIONAL RULES
RULE TERM_BY_TERM_DIFFERENTIATION:
  Within |x-a|<R, derivative of SUM c_n(x-a)^n is SUM(n=1..infinity) n*c_n(x-a)^(n-1).
RULE TERM_BY_TERM_INTEGRATION:
  Within |x-a|<R, integral from a to x equals SUM(n=0..infinity) c_n(x-a)^(n+1)/(n+1).
RULE RADIUS_PRESERVATION:
  Differentiation and integration preserve the radius R of a power series.
RULE LINEARITY:
  T_a[alpha*f+beta*g] = alpha*T_a[f]+beta*T_a[g] whenever derivatives exist.
RULE PRODUCT:
  Formal Taylor coefficients of fg satisfy h_n=SUM(k=0..n) f_k*g_(n-k).
RULE COMPOSITION:
  Formal composition is valid when the inner series has zero constant term; analytic composition also requires suitable domain conditions.
RULE UNIQUENESS:
  If a function has a convergent power-series representation near a, its coefficients are f^(n)(a)/n!.
RULE POLYNOMIAL_TERMINATION:
  Polynomial f of degree d -> f^(n)=0 for n>d and T_a[f]=f.
RULE SYMMETRY_EVEN:
  Even analytic f at a=0 -> odd Maclaurin coefficients vanish.
RULE SYMMETRY_ODD:
  Odd analytic f at a=0 -> even Maclaurin coefficients vanish.

8. CANONICAL EXAMPLES
EXAMPLE EXPONENTIAL:
  exp(x) = SUM(n=0..infinity) x^n/n!, R=infinity.
EXAMPLE SINE:
  sin(x) = SUM(n=0..infinity) (-1)^n*x^(2n+1)/(2n+1)!, R=infinity.
EXAMPLE COSINE:
  cos(x) = SUM(n=0..infinity) (-1)^n*x^(2n)/(2n)!, R=infinity.
EXAMPLE GEOMETRIC:
  1/(1-x) = SUM(n=0..infinity) x^n, |x|<1, R=1.
EXAMPLE LOGARITHM:
  log(1+x) = SUM(n=1..infinity) (-1)^(n+1)*x^n/n, -1<x<=1 for real x.
EXAMPLE ARCTANGENT:
  arctan(x) = SUM(n=0..infinity) (-1)^n*x^(2n+1)/(2n+1), |x|<=1 for real x.
EXAMPLE SMOOTH_NONANALYTIC:
  f(x)=exp(-1/x^2) for x!=0 and f(0)=0 has all derivatives zero at 0; T_0[f]=0 yet f(x)>0 for x!=0.

9. INFERENCE ENGINE
INFER 01: HasAllDerivativesAt(f,a) -> TaylorCoefficientsDefined(f,a).
INFER 02: TaylorCoefficientsDefined(f,a) -> FormalTaylorSeriesDefined(f,a).
INFER 03: TaylorSeriesConvergesAt(f,a,x) AND RemainderTendsToZero(f,a,x) -> TaylorSeriesRepresents(f,x).
INFER 04: AnalyticAt(f,a) -> ExistsLocalTaylorRepresentation(f,a).
INFER 05: SmoothAt(f,a) -/-> AnalyticAt(f,a).
INFER 06: Distance(x,a)<Radius(T) -> AbsolutelyConvergent(T,x).
INFER 07: Distance(x,a)>Radius(T) -> Divergent(T,x).
INFER 08: Distance(x,a)=Radius(T) -> BoundaryTestRequired(T,x).
INFER 09: Polynomial(f) -> ExactFiniteTaylorExpansion(f).
INFER 10: KnownDerivativeBound(f,N+1,M) -> LagrangeErrorBound(f,N,M).
INFER 11: AnalyticAt(f,a) AND AnalyticAt(g,a) -> AnalyticAt(f+g,a).
INFER 12: AnalyticAt(f,a) AND AnalyticAt(g,a) -> AnalyticAt(f*g,a).
INFER 13: ValidInteriorPoint(T,x) -> DifferentiationAllowed(T,x).
INFER 14: ValidInteriorPoint(T,x) -> IntegrationAllowed(T,x).
INFER 15: TaylorSeriesAtZero(T) -> MaclaurinSeries(T).
INFER 16: EqualityOnNeighbourhood(f,T) -> CoefficientUniqueness(f,T).

10. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE:
  Taylor expansion ~= Taylor series expansion [context: infinite expansion at a]
  Maclaurin expansion == Taylor expansion about zero
  Taylor polynomial ~= truncated Taylor series [finite order]
  Power series != necessarily Taylor representation of a given function
  Convergence of Taylor series != equality to originating smooth function
  Analytic != merely infinitely differentiable (over real variables)

11. VALIDATION AND CONSTRAINTS
CONSTRAINT centre a must lie in domain where all required derivatives exist.
CONSTRAINT n is integer >=0; n! is factorial.
CONSTRAINT convergence is defined through limits of partial sums.
CONSTRAINT no convergence claim on |x-a|=R without a separate test.
CONSTRAINT distinguish real interval from complex disk.
CONSTRAINT record conditions for each error formula.
CONSTRAINT formal coefficient computation alone is not proof of function equality.
CONSTRAINT approximation error claims require an applicable bound or proof.

12. KNOWLEDGE GRAPH TRIPLES
(TaylorSeries, subClassOf, PowerSeries)
(MaclaurinSeries, subClassOf, TaylorSeries)
(TaylorSeries, expansionCentre, ExpansionCentre)
(TaylorSeries, coefficientRule, DerivativeAtCentreDividedByFactorial)
(TaylorSeries, convergenceDeterminedBy, PartialSums)
(TaylorSeries, radiusDeterminedBy, CauchyHadamard)
(TaylorPolynomial, truncates, TaylorSeries)
(TaylorPolynomial, errorTerm, Remainder)
(AnalyticFunction, locallyRepresentedBy, TaylorSeries)
(SmoothFunction, notNecessarily, AnalyticFunction)

13. CONFIDENCE
CONFIDENCE:
  formal_definitions: 1.00
  standard_theorems: 0.99
  canonical_examples: 0.99
  source_page_alignment: unverified
  rule: Never convert heuristic approximation into a theorem without assumptions.

14. FLOW_HASH
FLOW_HASH:
  algorithm: SHA-256
  canonicalisation: UTF-8; LF line endings; deterministic field ordering
  value: TO_BE_COMPUTED_AFTER_FINALISATION
  note: Do not fabricate a cryptographic digest.

15. UPDATE_MODEL
UPDATE_MODEL:
  version: 4.0
  update_trigger: mathematical correction | ontology expansion | source-page revision
  compatibility: preserve entity identifiers and typed relationship meanings
  provenance: preserve source URL, version, revision date and author attribution
  verification: rerun domain checks, theorem assumptions, examples and hash computation

16. AI_CONSUMPTION
AI_CONSUMPTION:
  parse_mode: typed-entity-and-rule
  target: semantic search | symbolic reasoning | knowledge graph | mathematics education
  required_fields: entity_type, expansion_centre, coefficient_formula, convergence_radius, domain, proof_assumptions
  reasoning_priority: definitions > proved theorems > validated examples > heuristic approximations
  safety_check: do not infer f=T_a[f] solely from infinite differentiability
  output_contract: label formal equality, pointwise convergence, absolute convergence and numerical approximation separately

END FLOW ALGEBRA