Source: https://guthlabs.ai/docs/instruments/power-series-radius-of-convergence/implementation/

instrument implementation / implementation-contract

# Power Series Radius of Convergence implementation

A source-bound implementation contract for Power Series Radius of Convergence; deployment and executable qualification remain separate.

Status: **VERIFIED_CLEAN**Lifecycle: **implementation-contract**

## Binding contract

This record describes what an executable surface must bind and validate. It does not claim that a public calculator, package, or production route is deployed.

**Record owner**: Guth Instruments

**Lifecycle**: verified-clean

**Evidence ID**: t4-033

**Effect**: Computation or source lookup only

## Input and validation requirements

- 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.

## Required corrections

No binding erratum is attached to this unique-name record. Implementers must still preserve the declared units, domain, and failure behavior.

## Inspectable interface fixture

The sample below is evidence for interface and validation tests, not a production implementation.

````
{
  "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."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/power-series-radius-of-convergence/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/power-series-radius-of-convergence/mechanism/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/power-series-radius-of-convergence/qualification/>)

## 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`