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Chapter 22
Taylor Series as Recursive Expansion

Calculus and Continuous Change
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 15 June 2026
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Flow Algebra 
flow "Taylor Series as Recursive Expansion"

# ---------- CONCEPT ----------
# A Taylor series reconstructs a function from its local derivatives — a recursive expansion of coherence around a point.
# Each derivative encodes how form changes at that boundary; summing them recovers the whole (within a radius of convergence).
# Analytic functions equal their Taylor series on the domain of convergence.

# ---------- STATES ----------
state ExpansionSystem(T):
  Function:
    description: "f ∈ C^∞ near a (analytic if series converges to f)."
  BasePoint:
    description: "Expansion centre a (Maclaurin: a = 0)."
  Coefficients:
    description: "c_k = f^{(k)}(a)/k! — local change data."
  PolynomialN:
    description: "T_N(x) = Σ_{k=0}^N c_k (x−a)^k — Nth-order local model."
  Remainder:
    description: "R_N(x) = f(x) − T_N(x) — truncation error."
  ConvergenceDomain:
    description: "Radius R where series converges; may equal distance to nearest singularity."

# ---------- PARAMETERS ----------
params:
  centre: a
  order: N
  radius: R
  variable: x
  smoothness: "C^{N+1} for Lagrange remainder; analytic for equality."
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Taylor series: f(x) = Σ_{k=0}^∞ f^{(k)}(a)/k! · (x−a)^k (if analytic)."
  - "Nth polynomial: T_N(x) = Σ_{k=0}^N f^{(k)}(a)/k! · (x−a)^k."
  - "Remainder (Lagrange): R_N(x) = f^{(N+1)}(ξ)/(N+1)! · (x−a)^{N+1}, for some ξ between a and x."
  - "Convergence tests (ratio/root) determine |x−a| < R."
  - "Termwise differentiation/integration preserves convergence inside |x−a| < R."
  - "If f analytic, limit N→∞ T_N(x) = f(x) within radius."

# ---------- TRANSFORMS ----------
transforms:
  ExpandAt(f,a):
    rule: "Compute c_k = f^{(k)}(a)/k! and form Σ c_k (x−a)^k."
  ShiftCentre(b):
    rule: "Re-expand about b: substitute a→b, recompute derivatives."
  DifferentiateSeries:
    rule: "d/dx Σ c_k (x−a)^k = Σ k c_k (x−a)^{k−1} (inside radius)."
  IntegrateSeries:
    rule: "∫ Σ c_k (x−a)^k dx = C + Σ c_k (x−a)^{k+1}/(k+1)."
  EstimateError(N, M, ρ):
    meaning: "If |f^{(N+1)}(t)| ≤ M on segment, and |x−a|=ρ, Lagrange bound |R_N(x)| ≤ M ρ^{N+1}/(N+1)!"
  ComposeIfSmall(g):
    rule: "If |g(x)−a| < R, substitute x→g(x) into series (composition within domain)."

# ---------- MONITORS ----------
monitors:
  RatioTest:
    measure: "L = limsup |c_{k+1}/c_k|"
    output: "R = 1/L (with L=0 ⇒ R=∞, L=∞ ⇒ R=0)."
  RootTest:
    measure: "R = 1 / limsup √[k]{|c_k|}"
    meaning: "Alternative radius estimate."
  RemainderBound:
    track: "|R_N(x)|"
    meaning: "Guarantees approximation accuracy at order N."
  StabilityUnderOps:
    meaning: "Check validity after termwise diff/integration inside radius."
  SingularityProbe:
    meaning: "Locate nearest singularity; predicts R."

# ---------- EQUATIONS ----------
equations:
  - "f(x) = \\sum_{k=0}^{\\infty} \\frac{f^{(k)}(a)}{k!} (x-a)^k"
  - "T_N(x) = \\sum_{k=0}^{N} \\frac{f^{(k)}(a)}{k!} (x-a)^k"
  - "R_N(x) = \\frac{f^{(N+1)}(\\xi)}{(N+1)!} (x-a)^{N+1}"
  - "\\text{Ratio test: } R = 1/\\limsup \\left| \\frac{c_{k+1}}{c_k} \\right|"
  - "\\text{Root test: } R = 1/\\limsup \\sqrt[k]{|c_k|}"
  - "e^x = \\sum_{k=0}^{\\infty} \\frac{x^k}{k!},\\; \\sin x = \\sum_{k=0}^{\\infty} (-1)^k \\frac{x^{2k+1}}{(2k+1)!},\\; \\cos x = \\sum_{k=0}^{\\infty} (-1)^k \\frac{x^{2k}}{(2k)!}"
  - "\\frac{1}{1-x} = \\sum_{k=0}^{\\infty} x^k,\\; |x|<1"

# ---------- NOTES ----------
notes:
  - "Taylor expansion is recursive synthesis: derivatives at a encode all local behaviour; summation reconstructs the whole."
  - "Accuracy grows with order when |x−a| is well within the radius; near the boundary, convergence slows."
  - "The radius is typically the distance to the nearest complex-plane singularity — coherence breaks at singular boundaries."
  - "Analyticity is the closure condition: only then does the infinite sum equal the function."
  - "In Flow Algebra, Taylor series is the archetype of local-to-global coherence: infinitesimal change data folded back into a global form."
Explanatory Notes

The Taylor series is the language of local coherence — a way of rebuilding a function entirely from its own boundary derivatives.   Each derivative describes how the function bends, twists, or accelerates at a single point; together, they unfold into a complete reconstruction of the function’s behaviour near that point.

It is the mathematics of recursion: local change rebuilding global form.

For a function f(x)f(x) that is smooth near a point aa, the series

\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x - a)^k \]

represents the infinite summation of its local behaviour.
Each term captures a higher-order response — the first derivative gives slope, the second gives curvature, and higher derivatives capture the finer harmonics of change.

When the series converges, the infinite sequence of local patterns closes perfectly back into the original function, forming a complete loop of coherence.

The Nth-order Taylor polynomial,

\[ T_N(x) = \sum_{k=0}^{N} \frac{f^{(k)}(a)}{k!}(x - a)^k \]

provides an approximation that grows more accurate as NN increases.
The remainder term measures what has yet to be included:

\[ R_N(x) = \frac{f^{(N+1)}(\xi)}{(N+1)!}(x - a)^{N+1} \]

for some ξ\xi between aa and xx.
This expresses the exact coherence error — the part of the structure still missing from the truncated expansion.

The region where the series converges, known as the radius of convergence, depends not on where the function is defined, but on where its coherence fails.   The radius RR is equal to the distance from the expansion point to the nearest singularity — a location where the function ceases to be analytic.
At that boundary, coherence breaks down: the derivatives grow without limit, and the recursive structure can no longer rebuild the whole.

Thus, every Taylor series carries within it a map of its own stability — a boundary of coherence beyond which the local recursion cannot reach.

When a function is analytic, its Taylor series is not just an approximation but an equality within its radius of convergence:   the infinite sum and the function become indistinguishable.   This property represents the highest form of mathematical coherence — a function entirely defined by its own derivatives, self-contained and self-reconstructing.

Common examples illustrate this perfect recursion:

\[ e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}, \qquad \sin x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}, \qquad \cos x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{(2k)!} \]

In each case, infinite change reproduces finite form — coherence expressed as self-similar expansion.

In Flow Algebra, the Taylor series is viewed as a recursive coherence operator.
It converts local derivatives into a self-closing structure — a dynamic feedback between form and motion.
Each term propagates from the last; the sum reconstitutes the original pattern.
It is mathematics as recursion: local change expanding outward until the whole is restored.

Through the Taylor series, analysis becomes synthesis:
the infinitely small rebuilds the infinitely whole,
and coherence, once divided into its derivatives, finds its way home.

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