Infinite Series Ontology

FLOW ALGEBRA
INFINITE SERIES ONTOLOGY
Version: 4.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/infinite-series-ontology-2.html
Authorship: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library
Status: independently constructed mathematical ontology; source-page content not verified

01. ONTOLOGY IDENTITY
ONTOLOGY_ID := mathematical_entities.sequence_and_series.infinite_series
ENTITY := InfiniteSeries
PARENT := Series
DOMAIN := MathematicalAnalysis
DEFINITION := An ordered formal sum of infinitely many terms, interpreted through its sequence of finite partial sums.
NOTATION := sum_{n=n0}^infinity a_n
TERM_SEQUENCE := (a_n)_{n>=n0}
PARTIAL_SUM := S_N = sum_{n=n0}^N a_n
SERIES_SUM := lim_{N->infinity} S_N, only if the limit exists finitely in the specified space.

02. TYPES
Index := integer n >= n0
Scalar := Real | Complex
TermSequence := Function(Index, Scalar)
PartialSumSequence := Function(Index, Scalar)
Series := (TermSequence, StartingIndex, AmbientSpace)
ConvergenceStatus := CONVERGENT | DIVERGENT | UNDETERMINED
DivergenceMode := OSCILLATORY | UNBOUNDED | OTHER_NONCONVERGENT
ProofStatus := PROVED | DISPROVED | UNKNOWN
TestOutcome := CONVERGENT | DIVERGENT | INCONCLUSIVE | NOT_APPLICABLE

03. TYPED RELATIONSHIPS
InfiniteSeries HAS_TERM_SEQUENCE TermSequence
InfiniteSeries HAS_PARTIAL_SUM_SEQUENCE PartialSumSequence
TermSequence GENERATES PartialSumSequence
PartialSumSequence DETERMINES ConvergenceStatus
ConvergentSeries HAS_SUM Scalar
AbsolutelyConvergentSeries SUBCLASS_OF ConvergentSeries
ConditionallyConvergentSeries SUBCLASS_OF ConvergentSeries
PowerSeries SUBCLASS_OF InfiniteSeries
GeometricSeries SUBCLASS_OF InfiniteSeries
AlternatingSeries SUBCLASS_OF InfiniteSeries
HarmonicSeries INSTANCE_OF DivergentSeries

04. CORE DEFINITIONS
CONVERGES(series, L) iff forall epsilon>0 exists N forall m>=N: |S_m-L|<epsilon.
DIVERGES(series) iff no finite L satisfies CONVERGES(series,L).
CAUCHY(series) iff forall epsilon>0 exists N forall m>n>=N: |sum_{k=n+1}^m a_k|<epsilon.
ABSOLUTELY_CONVERGES(series) iff sum |a_n| converges.
CONDITIONALLY_CONVERGES(series) iff series converges and sum |a_n| diverges.
TAIL(series,N) := sum_{n=N+1}^infinity a_n when convergent.
REMAINDER(series,N) := L-S_N when L exists.

05. INVARIANTS AND NECESSARY CONDITIONS
R01: CONVERGES(series,L) => lim a_n=0.
R02: lim a_n != 0 OR lim a_n does not exist => DIVERGES(series).
R03: lim a_n=0 DOES_NOT_IMPLY CONVERGES(series).
R04: CONVERGES(series,L) <=> CAUCHY(series) over R or C.
R05: ABSOLUTELY_CONVERGES(series) => CONVERGES(series).
R06: CONDITIONALLY_CONVERGES(series) => CONVERGES(series) AND NOT ABSOLUTELY_CONVERGES(series).
R07: Changing finitely many terms preserves convergence status, but may change the sum.
R08: Removing or adding finitely many terms preserves convergence status.
R09: If sum a_n=A and sum b_n=B, then sum(alpha*a_n+beta*b_n)=alpha*A+beta*B.
R10: Convergent series may be regrouped into consecutive finite blocks without changing sum, provided original term order is retained.
R11: Arbitrary rearrangement of absolutely convergent real or complex series preserves its sum.
R12: Conditionally convergent real series may change sum or fail to converge under rearrangement.
R13: A bounded partial-sum sequence need not converge.
R14: Nonnegative terms imply nondecreasing partial sums; convergence iff partial sums are bounded above.

06. CANONICAL FAMILIES
GEOMETRIC: sum_{n=0}^infinity a*r^n = a/(1-r) for |r|<1; if a!=0 and |r|>=1, divergent.
P_SERIES: sum_{n=1}^infinity 1/n^p converges iff p>1 for real p.
HARMONIC: sum_{n=1}^infinity 1/n diverges.
ALTERNATING_HARMONIC: sum_{n=1}^infinity (-1)^(n+1)/n = ln(2), conditionally convergent.
TELESCOPING: sum_{n=1}^N (b_n-b_{n+1}) = b_1-b_{N+1}; convergence if b_N has finite limit.
EXPONENTIAL: sum_{n=0}^infinity z^n/n! = exp(z), absolutely convergent for all complex z.

07. CONVERGENCE TESTS
T01 TERM_TEST: if a_n does not tend to zero => divergent; zero limit => inconclusive.
T02 COMPARISON: 0<=a_n<=b_n eventually; sum b_n convergent => sum a_n convergent.
T03 COMPARISON_DIVERGENCE: 0<=b_n<=a_n eventually; sum b_n divergent => sum a_n divergent.
T04 LIMIT_COMPARISON: a_n,b_n>0 eventually and lim a_n/b_n=c with 0<c<infinity => same convergence status.
T05 RATIO: L=lim |a_(n+1)/a_n| where defined; L<1 => absolute convergence; L>1 => divergence; L=1 => inconclusive.
T06 ROOT: L=limsup |a_n|^(1/n); L<1 => absolute convergence; L>1 => divergence; L=1 => inconclusive.
T07 INTEGRAL: f positive, continuous, decreasing eventually, a_n=f(n); sum and improper integral have same convergence status.
T08 ALTERNATING: a_n=(-1)^n b_n with b_n>=0, eventually nonincreasing, b_n->0 => convergence.
T09 DIRICHLET: bounded partial sums of a_n, b_n monotone tending to zero => sum a_n*b_n converges.
T10 ABEL: sum a_n converges and b_n monotone bounded => sum a_n*b_n converges.
T11 CAUCHY_CONDENSATION: a_n positive nonincreasing; sum a_n converges iff sum 2^k*a_(2^k) converges.
T12 CAUCHY_CRITERION: series converges iff all sufficiently late finite tails have arbitrarily small absolute value (complete scalar space).
TEST_POLICY: Preconditions must be verified before applying any test.
TEST_POLICY: An inconclusive test is not evidence of convergence or divergence.

08. POWER SERIES
POWER_SERIES := sum_{n=0}^infinity c_n*(x-x0)^n.
RADIUS := R in [0,infinity], possibly infinity.
INTERIOR := |x-x0|<R => absolute convergence.
EXTERIOR := |x-x0|>R => divergence.
BOUNDARY := |x-x0|=R requires individual analysis.
CAUCHY_HADAMARD := 1/R = limsup |c_n|^(1/n), with conventional extended-real interpretations.
INTERIOR_OPERATIONS := termwise differentiation and integration within the radius of convergence.

09. ERROR ESTIMATION
ALTERNATING_REMAINDER: if alternating-series test applies, |L-S_N|<=b_(N+1).
GEOMETRIC_REMAINDER: sum_{n=N+1}^infinity a*r^n = a*r^(N+1)/(1-r) for |r|<1.
ABSOLUTE_TAIL: |sum_{n=N+1}^infinity a_n|<=sum_{n=N+1}^infinity |a_n| when absolutely convergent.
ERROR_CERTIFICATE := (method, assumptions, N, bound, proof_reference).

10. KNOWLEDGE GRAPH TRIPLES
(InfiniteSeries, isA, MathematicalEntity)
(InfiniteSeries, isA, Series)
(InfiniteSeries, definedBy, PartialSumLimit)
(PartialSumLimit, yields, ConvergenceStatus)
(AbsoluteConvergence, implies, Convergence)
(ConditionalConvergence, requires, Convergence)
(ConditionalConvergence, excludes, AbsoluteConvergence)
(GeometricSeries, hasParameter, CommonRatio)
(PowerSeries, hasProperty, RadiusOfConvergence)
(ConvergenceTest, requires, Preconditions)

11. INFERENCE FLOW
INPUT: -expression, index_range, scalar_domain, assumptions.
STEP_01: Parse the summand and starting index.
STEP_02: Construct term sequence and partial sums.
STEP_03: Evaluate necessary term-limit condition.
STEP_04: Detect known canonical series and exact transformations.
STEP_05: Classify sign structure and monotonicity when justified.
STEP_06: Select applicable tests and verify preconditions.
STEP_07: Apply valid convergence/divergence deductions.
STEP_08: Determine absolute versus conditional convergence if applicable.
STEP_09: Compute sum only when derivation is justified.
STEP_10: Produce proof trace, confidence, and provenance.
OUTPUT: status, convergence_kind, sum_if_known, test_trace, uncertainty.

12. VALIDATION CONSTRAINTS
V01: Never assign a finite series sum without a convergent partial-sum limit.
V02: Distinguish sequence convergence from series convergence.
V03: Do not infer convergence from a_n->0 alone.
V04: Preserve index bounds and ambient scalar domain.
V05: Do not use ratio/root test at threshold one to force a decision.
V06: Do not apply nonnegative comparison to signed terms without suitable transformation.
V07: Record boundary-point checks for power series separately.
V08: Do not claim a conditionally convergent series is rearrangement invariant.
V09: Distinguish formal sums, divergent sums, and regularised values.
V10: Record theorem hypotheses and proof obligations.

13. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE("infinite sum", "infinite series") := CONTEXT_DEPENDENT.
SEMANTIC_EQUIVALENCE("series convergence", "convergence of partial sums") := TRUE.
SEMANTIC_EQUIVALENCE("absolute convergence", "convergence of absolute-value series") := TRUE.
SEMANTIC_EQUIVALENCE("conditional convergence", "convergent but not absolutely convergent") := TRUE.
SEMANTIC_EQUIVALENCE("summation notation", "proven numerical sum") := FALSE.
SEMANTIC_DIFFERENCE("sequence", "series") := ordered terms versus accumulated partial sums.
SEMANTIC_DIFFERENCE("infinite series", "finite series") := unbounded versus bounded index range.

14. WORKED EXAMPLES
E01: sum_{n=0}^infinity (1/2)^n -> CONVERGENT, ABSOLUTE, SUM=2.
E02: sum_{n=1}^infinity 1/n -> DIVERGENT; terms tend to zero.
E03: sum_{n=1}^infinity 1/n^2 -> CONVERGENT, ABSOLUTE, SUM=pi^2/6.
E04: sum_{n=1}^infinity (-1)^(n+1)/n -> CONVERGENT, CONDITIONAL, SUM=ln(2).
E05: sum_{n=0}^infinity (-1)^n -> DIVERGENT; partial sums oscillate.
E06: sum_{n=1}^infinity 1/[n(n+1)] -> CONVERGENT, SUM=1 by telescoping.
E07: sum_{n=0}^infinity x^n -> CONVERGENT iff |x|<1 (real/complex x).

15. CONFIDENCE
CONFIDENCE.PROVED := 1.00 when backed by a valid formal proof and verified assumptions.
CONFIDENCE.DERIVED := 0.95 when standard theorem and all hypotheses are established.
CONFIDENCE.HEURISTIC := 0.50 when numeric or pattern evidence only.
CONFIDENCE.UNKNOWN := 0.00 when no sufficient evidence.
CONFIDENCE_POLICY := Scores describe evidence status, not mathematical truth probabilities.

16. FLOW_HASH
FLOW_HASH.ALGORITHM := SHA-256
FLOW_HASH.INPUT := canonical UTF-8 bytes of the released ontology file.
FLOW_HASH.VALUE := COMPUTE_AT_PUBLICATION
FLOW_HASH.POLICY := Recompute whenever canonical content changes.

17. UPDATE_MODEL
VERSION := 4.0
UPDATED := 2026-10-13
UPDATE_TRIGGER := corrected theorem | changed semantic relation | expanded test | validation improvement.
UPDATE_PROCESS := propose -> verify -> review -> version -> hash -> publish.
BACKWARD_COMPATIBILITY := Preserve ontology ID and typed relations unless a documented migration is supplied.

18. AI_CONSUMPTION
AI_CONSUMPTION.MODE := structured_semantic_reasoning
AI_CONSUMPTION.INPUT := series expression + assumptions + domain + requested property.
AI_CONSUMPTION.OUTPUT := machine-readable classification + human-readable proof.
AI_CONSUMPTION.REQUIRE := hypothesis checking, source provenance, uncertainty marking.
AI_CONSUMPTION.PROHIBIT := treating heuristic tests as proof, fabricated sums, silent theorem assumptions.
AI_CONSUMPTION.LINKS := SeriesOntology | FiniteSeriesOntology | ConvergentSequenceOntology | DivergentSequenceOntology.

19. END
END FLOW ALGEBRA