{
  "title": "Sequence convergence explorer",
  "human_url": "https://happytails.ai/en/calculators/sequence-convergence-explorer/",
  "agent_url": "https://happytails.ai/agents/en/calculators/sequence-convergence-explorer/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/sequence-convergence-explorer.md",
  "content": "Sequence convergence explorer\nTabulate 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.\nHow to use Sequence convergence explorer\nType the general term, such as (2n^2 + 1)/(n^2 - 3).\nChoose how many terms to tabulate.\nRun the explorer and read the limit, the method, the monotonicity and the bounds.\nDownload the JSON report or the CSV of terms.\nHow can we improve this tool?\nReport a problem or share an idea.\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\nUseful next steps\nGraphing calculator\nFractions\nContinued fraction explorer\nTerms 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.\nLimits\nFractions 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).\nMonotone and bounded\nFor 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.\nEvidence versus proof\nWhen no exact method applies the limit is reported as not determined. The decimal tail of the table never decides the answer.",
  "content_format": "text/plain",
  "visibility": "public",
  "links": [
    {
      "name": "Graphing calculator",
      "url": "https://happytails.ai/en/calculators/graphing-calculator/"
    },
    {
      "name": "Fractions",
      "url": "https://happytails.ai/en/calculators/fractions/"
    },
    {
      "name": "Continued fraction explorer",
      "url": "https://happytails.ai/en/calculators/continued-fraction-explorer/"
    }
  ],
  "tool": {
    "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"
    }
  },
  "markdown": "# Sequence convergence explorer\n\n- Human page: https://happytails.ai/en/calculators/sequence-convergence-explorer/\n- Markdown: https://happytails.ai/en/calculators/sequence-convergence-explorer.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/sequence-convergence-explorer/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `sequence-convergence-explorer` |\n| Category | calculators |\n| Implementation | Registered implementation. Registration alone is not a production-quality guarantee. |\n| Browser execution | Use the visible action or interactive controls. |\n| Browser processing | Browser; see tool notes for network behavior. |\n| HTTP API | Not callable through HTTP. |\n| Browser-agent interface | run_current_tool |\n\n### Implementation notes\n\nTabulates exact terms and finds limits, monotonicity and bounds on your device, with decimal evidence kept separate.\n\n### Inputs and settings\n\nText input: General term.\n\nExample or placeholder shown in the interface (not a validated fixture):\n\n```text\n(2n^2 + 1)/(n^2 - 3)\n```\n\nUse n as the index, such as n/2^n or (1 + 1/n)^n. Up to 1,000 characters. Review the settings before running.\n\n| Setting | Meaning | Control type | Default | Constraints |\n| --- | --- | --- | --- | --- |\n| `terms` | Number of terms | number | \"20\" | min=1 max=500 step=1 |\n\n### Browser-agent workflow\n\n1. Open the human page in a browser that supports WebMCP.\n2. Discover the interface exposed by that page; use its actual schema.\n3. Select files on the page first when the tool requires files.\n4. Call `run_current_tool` with valid input and settings.\n5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links.\n\nCalls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution.\n\n### Planning specification\n\nTerms 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.\n\nThis describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support.\n\n### Complete machine-readable capability record\n\n```json\n{\n  \"id\": \"sequence-convergence-explorer\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"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.\",\n  \"implementation\": {\n    \"fields\": [\n      {\n        \"key\": \"terms\",\n        \"label\": \"Number of terms\",\n        \"value\": \"20\",\n        \"type\": \"number\",\n        \"min\": 1,\n        \"max\": 500,\n        \"step\": 1,\n        \"help\": \"How many terms to tabulate, from 1 to 500.\"\n      }\n    ],\n    \"presentation\": \"metrics\",\n    \"inputCodeDirection\": \"ltr\",\n    \"registrationFamily\": \"exact-math-discrete\",\n    \"inputMaxLength\": 1000,\n    \"inputErrorCode\": \"EXACT_MATH_DISCRETE_EXPRESSION_TOO_LONG\",\n    \"note\": \"Tabulates exact terms and finds limits, monotonicity and bounds on your device, with decimal evidence kept separate.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": true,\n      \"inputMaxLength\": 1000,\n      \"inputMode\": \"text\",\n      \"label\": \"General term\",\n      \"placeholder\": \"(2n^2 + 1)/(n^2 - 3)\",\n      \"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.\",\n      \"helpParts\": [\n        \"Use n as the index, such as n/2^n or (1 + 1/n)^n.\",\n        \"Up to 1,000 characters.\",\n        \"Review the settings before running.\"\n      ],\n      \"action\": null\n    },\n    \"mode\": \"explicit\",\n    \"interactive\": false\n  },\n  \"execution\": \"Browser interface\",\n  \"api\": null,\n  \"browser_agent\": {\n    \"supported\": true,\n    \"name\": \"run_current_tool\",\n    \"discovery\": \"WebMCP on the human page in a supporting browser\",\n    \"verification\": \"See audit/API-AUDIT.md; availability is not verification\"\n  }\n}\n```\n\n## Page guide\n\nTabulate 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.\n\n### How to use Sequence convergence explorer\n\n1. Type the general term, such as (2n^2 + 1)/(n^2 - 3).\n2. Choose how many terms to tabulate.\n3. Run the explorer and read the limit, the method, the monotonicity and the bounds.\n4. Download the JSON report or the CSV of terms.\n\nHow can we improve this tool?\n\nReport a problem or share an idea.\n\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\n\n## Useful next steps\n\n- [Graphing calculator](https://happytails.ai/en/calculators/graphing-calculator/) ([Markdown](https://happytails.ai/en/calculators/graphing-calculator.md))\n\n- [Fractions](https://happytails.ai/en/calculators/fractions/) ([Markdown](https://happytails.ai/en/calculators/fractions.md))\n\n- [Continued fraction explorer](https://happytails.ai/en/calculators/continued-fraction-explorer/) ([Markdown](https://happytails.ai/en/calculators/continued-fraction-explorer.md))\n\nTerms 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.\n\n## Limits\n\nFractions 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).\n\n### Monotone and bounded\n\nFor 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.\n\n### Evidence versus proof\n\nWhen no exact method applies the limit is reported as not determined. The decimal tail of the table never decides the answer.\n"
}