Series Convergence Tester mechanism

Model scope

The supplied ratio and root expressions are recognizable standard tests, but the specification omits their decision rules and omits the actual algorithms for three other claimed tests. The sum-estimation method is unspecified and therefore not reproducible or guaranteed. The honest label appropriately mentions heuristic selection and borderline cases, but it does not resolve these missing formulas, constraints, and error guarantees.

• The ratio-test formula is incomplete without the decision rule: L<1 implies absolute convergence, L>1 (or an infinite limit) implies divergence, and L=1 is inconclusive.
• The root-test formula is incomplete without the same decision rule and should explicitly use the limit of |a_n|^(1/n).
• No formulas or decision procedures are specified for the claimed comparison test, integral test, or alternating-series test.
• Partial-sum extrapolation is not defined: the spec gives no extrapolation model, truncation index, remainder estimate, stopping criterion, or error bound. Therefore a reliable sum estimate is not determined by the stated formulas.
• The ratio expression can divide by zero when a_n=0, and behavior at zero terms or eventually-zero terms is unspecified.
• A convergence verdict does not by itself provide a sum estimate; conditional convergence and slowly convergent series require explicit remainder handling.

Units and conventions

• Series index n is a positive integer, starting at n=1 unless otherwise specified.
• Terms a_n are dimensionless, or all terms have a common unit so that ratios and sums are meaningful; the spec does not state this.
• The ratio and root limits are mathematical limits as n approaches infinity, not finite-sample values.
• For the integral test, a_n must be represented by a positive, continuous, eventually decreasing function f(x) on a tail interval.
• For the alternating-series test, terms must have the form (-1)^n b_n or (-1)^(n+1)b_n, with b_n nonnegative, eventually decreasing, and tending to zero.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "general_term": "a_n=(1/3)^n",
    "n_start": 1,
    "test_selection": "ratio",
    "partial_sum_terms": 5
  },
  "expected_outputs": {
    "verdict": "convergent",
    "convergence_type": "absolute",
    "test_used": "ratio",
    "ratio_limit": "1/3",
    "partial_sum_S5": "121/243",
    "partial_sum_S5_decimal": 0.4979423868312757,
    "infinite_sum": "1/2",
    "infinite_sum_decimal": 0.5
  },
  "working": "a_1=1/3, a_2=1/9, a_3=1/27. First ratio: |a_2/a_1|=(1/9)/(1/3)=1/3. Second ratio: |a_3/a_2|=(1/27)/(1/9)=1/3; therefore L=1/3<1, so the series converges absolutely. S_5=1/3+1/9+1/27+1/81+1/243=(81+27+9+3+1)/243=121/243=0.4979423868312757. Using the geometric-series sum, S=(1/3)/(1-1/3)=(1/3)/(2/3)=1/2=0.5."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The ratio-test path has an explicit 0/0 trap for the identically zero series and must reject or reroute that input. Boundary cases with ratio limit 1 also require a different test, and a sign/indexing convention changes the reported sum for alternating series.

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-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json