happytails.ai

Sequence convergence explorer

Open Sequence convergence explorer on Happytails.ai. This view has the same page content and links, with explicit tool capabilities. Read the processing notes before using a tool.

Tool capabilities

{
  "id": "sequence-convergence-explorer",
  "category": "calculators",
  "built": true,
  "requirements": "Terms are limited to 1,000 characters and one index variable; 1 to 500 terms are tabulated, exactly while values stay under 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of terms.",
  "implementation": {
    "fields": [
      {
        "key": "terms",
        "label": "Number of terms",
        "value": "20",
        "type": "number",
        "min": 1,
        "max": 500,
        "step": 1,
        "help": "How many terms to tabulate, from 1 to 500."
      }
    ],
    "presentation": "metrics",
    "inputCodeDirection": "ltr",
    "registrationFamily": "exact-math-discrete",
    "inputMaxLength": 1000,
    "inputErrorCode": "EXACT_MATH_DISCRETE_EXPRESSION_TOO_LONG",
    "note": "Tabulates exact terms and finds limits, monotonicity and bounds on your device, with decimal evidence kept separate.",
    "experience": {
      "automatic": false,
      "single": true,
      "inputMaxLength": 1000,
      "inputMode": "text",
      "label": "General term",
      "placeholder": "(2n^2 + 1)/(n^2 - 3)",
      "help": "Use n as the index, such as n/2^n or (1 + 1/n)^n. Up to 1,000 characters. Review the settings before running.",
      "helpParts": [
        "Use n as the index, such as n/2^n or (1 + 1/n)^n.",
        "Up to 1,000 characters.",
        "Review the settings before running."
      ],
      "action": null
    },
    "mode": "explicit",
    "interactive": false
  },
  "execution": "Browser interface",
  "api": null,
  "browser_agent": {
    "supported": true,
    "name": "run_current_tool",
    "discovery": "WebMCP on the human page in a supporting browser",
    "verification": "See audit/API-AUDIT.md; availability is not verification"
  }
}

Page guide

Tabulate the terms of a sequence exactly and decide its limit, monotonicity and bounds where an exact method applies. Decimal tails are shown separately as evidence.

How to use Sequence convergence explorer
  1. Type the general term, such as (2n^2 + 1)/(n^2 - 3).
  2. Choose how many terms to tabulate.
  3. Run the explorer and read the limit, the method, the monotonicity and the bounds.
  4. Download the JSON report or the CSV of terms.

Useful next steps

Graphing calculatorFractionsContinued fraction explorer

Terms are limited to 1,000 characters and one index variable; 1 to 500 terms are tabulated, exactly while values stay under 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of terms.

Limits

Fractions of polynomials are decided by comparing degrees. Expressions with roots use the exact leading term c*n^p, including the conjugate trick for sqrt(n^2 + n) - n. Geometric factors such as n/2^n, the limit of (1 + a/n)^(b n) = exp(a b), n^(1/n) and bounded factors such as sin(n)/n are recognised. Method: OpenStax, Calculus Volume 2, section 5.1 Sequences (2016).

Monotone and bounded

For fractions of polynomials the sign of a(n+1) - a(n) is constant beyond its last real root, which is found exactly; the earlier terms are compared exactly to find where the pattern starts. Bounds then follow from the first terms and the limit.

Evidence versus proof

When no exact method applies the limit is reported as not determined. The decimal tail of the table never decides the answer.