Power Series Radius of Convergence mechanism

Model scope

Core radius formula passes under the Cauchy–Hadamard interpretation, but the specification needs review because the ratio-test wording is mathematically imprecise, the real-versus-complex domain is unspecified, the coefficient-index convention is unspecified, and arbitrary coefficient rules cannot guarantee automated endpoint classification. The stated honest label appropriately warns that endpoint convergence requires manual case checking.

• R=1/limsup|cₙ|^(1/n) is the correct Cauchy–Hadamard formula, with extended-value conventions, but it is not itself the ratio-test formula. A ratio-test formula requires an additional limit such as R=lim|cₙ/cₙ₊₁| when that limit exists.
• The claimed interval output cannot be fully determined from the radius formula alone; both endpoint cases require separate convergence tests. The honest label correctly states this limitation.
• The coefficient index convention is unspecified: whether the series starts at n=0 or n=1 affects examples and the constant term, although not usually the radius.
• The input specification does not state whether x is restricted to real values (an interval) or allowed to be complex (a disk of convergence).

Units and conventions

• x and center a are expressed in the same units, so x−a and R have those units.
• If x carries physical units, cₙ must carry reciprocal units sufficient to make each term cₙ(x−a)ⁿ dimensionally consistent; otherwise the root expression is not dimensionally meaningful without a normalized variable.
• For real x, the convergence set inside the radius is an interval; for complex x, it is a disk, with boundary behavior handled pointwise.
• The limsup is taken as n tends to infinity over the stated coefficient index set.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "coefficient_rule": "cₙ=2ⁿ for n≥0",
    "center_a": 3
  },
  "expected_outputs": {
    "radius_R": 0.5,
    "interval": "(2.5, 3.5)",
    "endpoint_behavior": "Both endpoints diverge"
  },
  "working": "|cₙ|^(1/n)=|2ⁿ|^(1/n)=2. Therefore limsup|cₙ|^(1/n)=2 and R=1/2=0.5. The open radius interval is 3−0.5<x<3+0.5, so 2.5<x<3.5. At x=2.5, cₙ(x−a)ⁿ=2ⁿ(−0.5)ⁿ=(−1)ⁿ, whose terms do not approach 0. At x=3.5, the terms are 2ⁿ(0.5)ⁿ=1, also not approaching 0; both endpoints diverge."
}

Failure boundary

• Division by zero occurs when limsup|cₙ|^(1/n)=0; this must be handled as R=∞, not as an error.
• If the limsup is +∞, the radius is R=0; this convention must be implemented explicitly.
• Endpoint convergence cannot be inferred from R and must be tested separately; the terms must at least tend to zero.
• A ratio-test implementation can fail when cₙ=0, when coefficients change sign, or when the coefficient ratio has no limit; the root-test limsup formula remains the general method.
• The implementation must distinguish the numerical value 0 from missing, malformed, or non-finite coefficients.
• The output label 'interval' is incorrect or ambiguous for complex inputs, where the corresponding object is a disk.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json