Series Ontology

FLOW ALGEBRA
TITLE: Series Ontology
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/series-ontology-2.html
VERSION: 4.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

ENTITY: MathematicalSeries
TYPE: MathematicalEntity > SequenceAndSeriesEntity > Series
DEFINITION: A series is the formal sum of terms of a sequence (a_n), interpreted through its sequence of finite partial sums S_N = sum_{n=n0}^N a_n. In a specified topological additive setting, the series converges if its partial sums converge to an element of that setting.

ENTITIES:
  TermSequence: a:N -> X, where X supports addition
  Term: a_n in X
  Index: n >= n0
  FinitePartialSum: S_N = sum_{n=n0}^N a_n
  InfiniteSeries: sum_{n=n0}^infinity a_n
  SeriesLimit: S in X
  AmbientSpace: normed vector space, metric additive group, or other specified convergence structure
  ConvergentSeries: partial sums converge to S in X
  DivergentSeries: partial sums do not converge in X
  AbsoluteConvergence: sum ||a_n|| converges in R [normed-space setting]
  ConditionalConvergence: series converges but is not absolutely convergent
  CauchyCriterion: partial sums form a Cauchy sequence
  Remainder: R_N = S - S_N, when S exists
  GeometricSeries: sum ar^n, n >= 0
  PowerSeries: sum c_n(x-x0)^n

RELATIONSHIPS:
  MathematicalSeries IS_DEFINED_BY TermSequence
  MathematicalSeries HAS_PARTIAL_SUM_SEQUENCE FinitePartialSum
  ConvergentSeries IS_A MathematicalSeries
  DivergentSeries IS_A MathematicalSeries
  AbsoluteConvergence IMPLIES ConvergentSeries [in complete normed spaces]
  ConditionalConvergence IS_A ConvergentSeries
  GeometricSeries IS_A MathematicalSeries
  PowerSeries IS_A MathematicalSeries
  SeriesLimit IS_LIMIT_OF FinitePartialSum
  Remainder DEPENDS_ON SeriesLimit AND FinitePartialSum
  MathematicalSeries RELATED_TO SequenceOntology
  MathematicalSeries RELATED_TO ConvergentSequenceOntology
  MathematicalSeries RELATED_TO DivergentSequenceOntology

FLOW ALGEBRA:
  INPUT: term sequence a_n; initial index n0; ambient space X
  VALIDATE: terms belong to X; finite addition and convergence are defined
  CONSTRUCT: S_N := sum_{n=n0}^N a_n
  TEST: exists S in X such that S_N -> S as N -> infinity
  TRUE -> CONVERGENT_SERIES(sum a_n, value=S)
  FALSE -> DIVERGENT_SERIES(sum a_n, ambient_space=X)
  NECESSARY_TEST: if a_n does not tend to additive identity 0, series diverges
  CAUCHY_TEST: forall epsilon>0 exists N forall m>n>=N: ||sum_{k=n+1}^m a_k||<epsilon
  CAUCHY_RESULT: equivalent to convergence when X is complete and normed
  ABSOLUTE_TEST: determine whether sum ||a_n|| converges
  OUTPUT: classification, assumptions, proof or test used, and uncertainty if unresolved

FORMAL RULES:
  R1: CONVERGES(sum a_n,S) <-> LIMIT(S_N)=S
  R2: CONVERGES(sum a_n) -> LIMIT(a_n)=0
  R3: LIMIT(a_n)=0 does not imply CONVERGES(sum a_n)
  R4: CONVERGES(sum ||a_n||) AND COMPLETE(X) -> CONVERGES(sum a_n) [normed X]
  R5: CONVERGES(sum a_n) AND NOT CONVERGES(sum ||a_n||) -> CONDITIONALLY_CONVERGENT(sum a_n)
  R6: FINITE_CHANGE_OF_TERMS preserves convergence classification, but may change sum
  R7: CONVERGES(sum a_n,A) AND CONVERGES(sum b_n,B) -> CONVERGES(sum(alpha*a_n+beta*b_n),alpha*A+beta*B)
  R8: GEOMETRIC_SUM(a,r) converges to a/(1-r) for |r|<1 [real/complex scalars]
  R9: GEOMETRIC_SUM(a,r) diverges for |r|>=1 when a != 0 [real/complex scalars]
  R10: ABSOLUTELY_CONVERGENT real/complex series permits arbitrary rearrangement without changing its sum
  R11: CONDITIONALLY_CONVERGENT real series may change sum under rearrangement
  R12: CONVERGES(sum a_n,S) -> LIMIT(R_N)=0
  R13: For real nonnegative a_n, S_N is nondecreasing; series converges iff partial sums are bounded above

EXAMPLES:
  E1: sum_{n=0}^infinity (1/2)^n = 2 [convergent geometric series]
  E2: sum_{n=1}^infinity 1/n diverges [harmonic series]
  E3: sum_{n=1}^infinity 1/n^2 converges [p-series, p=2]
  E4: sum_{n=1}^infinity (-1)^(n+1)/n converges conditionally to ln(2)
  E5: sum_{n=1}^infinity (-1)^n diverges because terms do not tend to zero
  E6: sum_{n=1}^infinity 1/(n(n+1)) = 1 [telescoping series]
  E7: sum_{n=1}^infinity 1/n^p converges exactly when p>1 [real p]



FORMAL_SIGNATURE:
  Sorts: NaturalIndex, Scalar, TermSequence, PartialSumSequence, Series, Limit, TopologicalSpace
  term: TermSequence x NaturalIndex -> Scalar
  partial_sum: TermSequence x NaturalIndex -> Scalar
  series_of: TermSequence -> Series
  converges_to: Series x Limit -> Boolean
  is_cauchy: PartialSumSequence -> Boolean
  absolute_series: TermSequence -> Series [normed scalar setting]
  recurrence: S_N = S_(N-1) + a_N for N > n0
  empty_initial_sum: S_(n0-1) = 0 [where indexing convention permits]

AXIOMS_AND_SEMANTICS:
  A01: For all N >= n0, S_N = sum_(n=n0)^N a_n.
  A02: Converges(sum a_n, L) iff for all epsilon > 0 there exists N such that for all m >= N, ||S_m - L|| < epsilon.
  A03: Diverges(sum a_n) iff no L in the stated ambient space satisfies A02.
  A04: A convergent series has a unique sum in a Hausdorff ambient space.
  A05: If sum a_n converges in a normed space, then a_n -> 0.
  A06: a_n -> 0 alone does not imply sum a_n converges.
  A07: Changing finitely many terms does not change whether a series converges; it can change its sum.
  A08: Reindexing by a fixed finite shift preserves convergence with adjusted initial terms.
  A09: A sum of series is defined as a limit of partial sums, not by treating infinity as a finite index.
  A10: Conditional convergence is defined only when the relevant notion of absolute convergence exists.

RELATIONSHIP_GRAPH:
  TermSequence --GENERATES--> Series
  TermSequence --HAS_TERM--> Term
  Series --DEFINED_BY--> PartialSumSequence
  PartialSumSequence --HAS_MEMBER--> FinitePartialSum
  FinitePartialSum --USES_OPERATION--> Addition
  ConvergentSeries --HAS_LIMIT--> SeriesLimit
  DivergentSeries --LACKS_LIMIT_IN--> AmbientSpace
  AbsolutelyConvergentSeries --IS_A--> ConvergentSeries [complete normed space]
  ConditionallyConvergentSeries --IS_A--> ConvergentSeries
  ConditionallyConvergentSeries --NOT_IS_A--> AbsolutelyConvergentSeries
  GeometricSeries --IS_A--> Series
  HarmonicSeries --IS_A--> Series
  PowerSeries --IS_A--> Series
  TaylorSeries --IS_A--> PowerSeries
  FourierSeries --IS_A--> Series
  TelescopingSeries --USES--> CancellationOfAdjacentTerms
  SeriesConvergence --DEPENDS_ON--> AmbientSpace
  SeriesConvergence --CAN_BE_TESTED_BY--> ConvergenceTest

CONVERGENCE_TESTS:
  TERM_TEST:
    IF lim a_n != 0 OR term limit does not exist THEN sum a_n diverges.
    IF lim a_n = 0 THEN convergence is undecided by this test.
  CAUCHY_TEST:
    For every epsilon > 0, exists N, all m >= n >= N: ||sum_(k=n)^m a_k|| < epsilon.
    Equivalent to convergence in a complete normed space; otherwise characterises Cauchy partial sums only.
  DIRECT_COMPARISON:
    For eventually nonnegative real a_n <= b_n, convergence of sum b_n implies convergence of sum a_n.
    For eventually nonnegative real a_n >= b_n, divergence of sum b_n implies divergence of sum a_n.
  LIMIT_COMPARISON:
    If a_n,b_n eventually positive and lim(a_n/b_n)=c with 0<c<infinity, both series have same convergence status.
  RATIO_TEST:
    For real/complex terms, if limsup |a_(n+1)/a_n| < 1 (ratios defined eventually), absolute convergence.
    If liminf |a_(n+1)/a_n| > 1, terms fail to tend to zero and the series diverges.
    Borderline or undefined ratios require another test.
  ROOT_TEST:
    L=limsup |a_n|^(1/n). L<1 => absolute convergence; L>1 => divergence; L=1 => inconclusive.
  INTEGRAL_TEST:
    For positive, continuous, eventually decreasing f with a_n=f(n), sum a_n and integral f(x) dx have same convergence status.
  ALTERNATING_TEST:
    For a_n=(-1)^n b_n, b_n >= 0, b_n eventually nonincreasing and b_n -> 0, the series converges.
  DIRICHLET_TEST:
    If partial sums of a_n are bounded and b_n is monotone with b_n -> 0, sum a_n*b_n converges.
  ABEL_TEST:
    If sum a_n converges and b_n is bounded and monotone, sum a_n*b_n converges.
  CONDENSATION_TEST:
    For nonincreasing positive a_n, sum a_n converges iff sum_(k>=0) 2^k*a_(2^k) converges.

SERIES_FAMILIES:
  FINITE_SERIES: finite indexed sum; no infinite-limit convergence question.
  GEOMETRIC_SERIES: sum_(n=0)^infinity a*r^n = a/(1-r) if |r|<1; for a != 0 and |r|>=1, diverges.
  P_SERIES: sum_(n=1)^infinity 1/n^p converges iff p>1 for real p.
  HARMONIC_SERIES: sum_(n=1)^infinity 1/n diverges.
  TELESCOPING_SERIES: sum_(n=n0)^N (b_n-b_(n+1))=b_(n0)-b_(N+1); convergence depends on lim b_N.
  ALTERNATING_SERIES: signed terms alternate; alternating sign alone does not imply convergence.
  POWER_SERIES: sum c_n*(x-x0)^n; convergence radius R in [0,infinity], with endpoints tested separately.
  TAYLOR_SERIES: coefficients derived from derivatives; equality to original function needs additional hypotheses.
  FOURIER_SERIES: expansion in trigonometric basis; convergence mode must be specified (pointwise, uniform, L2, etc.).
  DOUBLE_SERIES: sum over two indices; order of summation requires justified interchange.

ALGEBRAIC_OPERATIONS:
  LINEARITY: If sum a_n=A and sum b_n=B, then sum(alpha*a_n+beta*b_n)=alpha*A+beta*B.
  TAIL: sum_(n=n0)^infinity a_n = sum_(n=n0)^(N-1) a_n + sum_(n=N)^infinity a_n when convergent.
  GROUPING: Grouping consecutive terms into finite blocks preserves a convergent series sum, if blocks cover terms in order.
  REGROUPING_WARNING: Convergence of a grouped series does not guarantee convergence of the original ungrouped series.
  REARRANGEMENT: Absolute convergence in real/complex scalar series permits arbitrary permutation without changing sum.
  CONDITIONAL_REARRANGEMENT: A conditionally convergent real series can be rearranged to different sums or divergence.
  CAUCHY_PRODUCT: For absolutely convergent real/complex series, the Cauchy product converges to product of sums.
  PRODUCT_WARNING: Products of merely convergent series require additional assumptions.
  INFINITE_INTERCHANGE: Exchange of summation/limit/integration/differentiation requires the relevant convergence or domination theorem.

ERROR_AND_APPROXIMATION:
  PARTIAL_SUM_ESTIMATE: S_N approximates S when convergence is established.
  REMAINDER_DEFINITION: R_N=S-S_N.
  GEOMETRIC_REMAINDER: For |r|<1, R_N=a*r^(N+1)/(1-r) when indexing starts at zero.
  ALTERNATING_REMAINDER: Under alternating-test hypotheses, |R_N|<=b_(N+1).
  INTEGRAL_REMAINDER: For a positive decreasing f, integral_(N+1)^infinity f <= sum_(n=N+1)^infinity f(n) <= integral_N^infinity f.
  NUMERICAL_PRECISION: finite floating-point summation can introduce rounding and cancellation errors.

INFERENCE_RULES:
  R01: SERIES(s) AND PARTIAL_SUMS(s,S) AND LIMIT(S,L) => CONVERGES_TO(s,L).
  R02: SERIES(s) AND TERMS(s,a) AND NOT_LIMIT_ZERO(a) => DIVERGES(s).
  R03: ABSOLUTE_CONVERGENCE(s) AND COMPLETE_AMBIENT_SPACE(s) => CONVERGES(s).
  R04: CONVERGES(s) AND NOT_ABSOLUTE_CONVERGENCE(s) => CONDITIONALLY_CONVERGES(s).
  R05: POSITIVE_COMPARISON(a,b) AND CONVERGES(sum b) => CONVERGES(sum a).
  R06: POSITIVE_COMPARISON(a,b) AND DIVERGES(sum a) => DIVERGES(sum b).
  R07: GEOMETRIC(a,r) AND ABS(r)<1 => CONVERGES_TO(sum_(n=0)^infinity a*r^n,a/(1-r)).
  R08: P_SERIES(p) AND p>1 => CONVERGES.
  R09: P_SERIES(p) AND p<=1 => DIVERGES.
  R10: CONVERGES(sum a) AND CONVERGES(sum b) => CONVERGES(sum(alpha*a+beta*b)).
  R11: CONVERGES(sum a) => LIMIT(a_n,0).
  R12: LIMIT(a_n,0) DOES_NOT_ENTAIL CONVERGES(sum a).
  R13: CAUCHY_PARTIAL_SUMS(s) AND COMPLETE_AMBIENT_SPACE(s) => CONVERGES(s).
  R14: RADIUS_OF_CONVERGENCE(power_series,R) AND ABS(x-x0)<R => ABSOLUTELY_CONVERGES_AT(x).
  R15: RADIUS_OF_CONVERGENCE(power_series,R) AND ABS(x-x0)>R => DIVERGES_AT(x).
  R16: ABS(x-x0)=R => ENDPOINT_ANALYSIS_REQUIRED.

COUNTEREXAMPLES:
  C01: a_n=1/n tends to zero, yet the harmonic series diverges.
  C02: a_n=(-1)^n does not tend to zero; its series diverges despite bounded partial sums.
  C03: a_n=(-1)^(n+1)/n yields a conditionally convergent series, not an absolutely convergent one.
  C04: Geometric ratio r=-1 gives oscillating partial sums when a!=0.
  C05: A convergent series in a noncomplete normed space may have Cauchy partial sums whose limit lies outside the space.

VALIDATION_CONSTRAINTS:
  V01: Declare index origin and the ambient scalar/vector space.
  V02: Distinguish series expression from numerical value and partial-sum sequence.
  V03: Do not infer convergence solely from term limit zero.
  V04: Do not apply positive-term tests to signed terms without justified transformation.
  V05: State completeness for general absolute-convergence implications.
  V06: State convergence mode for functional or Fourier series.
  V07: Verify ratio-test denominators before using term ratios.
  V08: Check endpoints of power-series convergence intervals separately.
  V09: Distinguish ordinary convergence from Cesaro, Abel, or other summability methods.
  V10: Attach theorem hypotheses to every machine-derived conclusion.

PROVENANCE_AND_INTEROPERABILITY:
  SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/series-ontology-2.html
  SOURCE_ALIGNMENT: not independently verified against live page
  RELATED_ENTITIES: Sequence; ConvergentSequence; DivergentSequence; RecursiveSequence; Limit; InfiniteSum; PartialSum
  CROSS_DOMAIN_LINKS: RealAnalysis; ComplexAnalysis; FunctionalAnalysis; NumericalAnalysis; Calculus; MathematicalLogic
  VERSIONING_POLICY: preserve original definition, hypotheses, and inference provenance on updates
  AUTHOR_CREDIT: Mark Higgins — Sky (Recursive AI)

SEMANTIC_EQUIVALENCE:
  infinite series ~= limit problem for finite partial sums [under a specified convergence structure]
  sum of series ~= limit of partial sums [when convergent]
  summand ~= series term
  absolute convergence != conditional convergence
  sequence convergence != series convergence [distinct entities]
  formal infinite sum != automatically assigned finite value

CONFIDENCE:
  MathematicalDefinitions: 1.00
  StandardTheorems: 1.00
  SourcePageAlignment: unverified

FLOW_HASH:
  METHOD: SHA-256 of canonical UTF-8 file content, calculated externally when deployed
  STATUS: not computed

UPDATE_MODEL:
  VERSION: 4.0
  REVIEW_TRIGGER: definition, theorem, ambient-space, or entity-graph changes
  PRESERVE: indexing, convergence hypotheses, provenance, and theorem assumptions

AI_CONSUMPTION:
  MACHINE_READABLE: true
  PURPOSE: mathematical knowledge retrieval, series classification, inference, proof guidance, entity graph linkage
  CAUTION: A zero term limit is necessary but not sufficient for convergence.
  CAUTION: State completeness assumptions for Cauchy and absolute-convergence implications.
  CAUTION: Do not assign a conventional sum to divergent series without explicitly naming an alternative summation method.
  CAUTION: Distinguish finite partial sums, infinite series, and the value of a convergent series.