{
  "title": "Series convergence checker",
  "human_url": "https://happytails.ai/en/calculators/series-convergence-checker/",
  "agent_url": "https://happytails.ai/agents/en/calculators/series-convergence-checker/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/series-convergence-checker.md",
  "content": "Series convergence checker\nApply supported convergence tests after checking the original term domain at every included integer index. The report distinguishes a proved verdict from an inconclusive result; supported geometric series also get an exact sum.\nHow to use Series convergence checker\nType the general term, such as 3/2^n, or set the start with a prefix such as n=2: 1/(n*ln(n)^2).\nRun the checker to apply the tests in order.\nRead the verdict, the deciding test and the preconditions of every test.\nDownload the JSON report or the CSV files of tests and partial sums.\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\nScientific calculator\nTerms are limited to 1,000 characters and one index variable. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests.\nTests used\nThe checker applies the nth-term test, the geometric series test, the ratio and root tests, limit comparison with a p-series using the exact leading term c*n^p, direct comparison for bounded factors such as sin(n), the alternating series test with exact checks that the terms decrease to zero, and the integral test for 1/(n*ln(n)^p). Method: OpenStax, Calculus Volume 2, sections 5.2 to 5.6 (2016).\nPreconditions\nOriginal denominators and other supported domain conditions are checked on the entire included integer tail before cancellation or zero-factor simplification. An included undefined term is rejected even when it occurs beyond the partial-sum table. When the full domain cannot be proved by the supported method, the checker returns a recoverable error. A test decides a verdict only when its preconditions are proved; otherwise the verdict remains inconclusive.\nPartial sums\nPartial sums are listed for orientation. They are exact where the term evaluator returns exact rational values and otherwise use decimals, even when the terms are mathematically rational. They never prove convergence or divergence.",
  "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": "Scientific calculator",
      "url": "https://happytails.ai/en/calculators/scientific-calculator/"
    }
  ],
  "tool": {
    "id": "series-convergence-checker",
    "category": "calculators",
    "built": true,
    "requirements": "Terms are limited to 1,000 characters and one index variable. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests.",
    "implementation": {
      "fields": [],
      "presentation": "metrics",
      "inputCodeDirection": "ltr",
      "registrationFamily": "exact-math-discrete",
      "inputMaxLength": 1000,
      "inputErrorCode": "EXACT_MATH_DISCRETE_EXPRESSION_TOO_LONG",
      "note": "Applies the classical convergence tests with their preconditions on your device and shows partial sums separately.",
      "experience": {
        "automatic": false,
        "single": true,
        "inputMaxLength": 1000,
        "inputMode": "text",
        "label": "General term",
        "placeholder": "3/2^n",
        "help": "Use n as the index. Add a start such as n=2: before the term if needed. Up to 1,000 characters. Review the settings before running.",
        "helpParts": [
          "Use n as the index. Add a start such as n=2: before the term if needed.",
          "Up to 1,000 characters.",
          "Review the settings before running."
        ],
        "action": null
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    "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": "# Series convergence checker\n\n- Human page: https://happytails.ai/en/calculators/series-convergence-checker/\n- Markdown: https://happytails.ai/en/calculators/series-convergence-checker.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/series-convergence-checker/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `series-convergence-checker` |\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\nApplies the classical convergence tests with their preconditions on your device and shows partial sums separately.\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\n3/2^n\n```\n\nUse n as the index. Add a start such as n=2: before the term if needed. Up to 1,000 characters. Review the settings before running.\n\nNo additional settings are declared for this tool.\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. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests.\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\": \"series-convergence-checker\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"requirements\": \"Terms are limited to 1,000 characters and one index variable. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests.\",\n  \"implementation\": {\n    \"fields\": [],\n    \"presentation\": \"metrics\",\n    \"inputCodeDirection\": \"ltr\",\n    \"registrationFamily\": \"exact-math-discrete\",\n    \"inputMaxLength\": 1000,\n    \"inputErrorCode\": \"EXACT_MATH_DISCRETE_EXPRESSION_TOO_LONG\",\n    \"note\": \"Applies the classical convergence tests with their preconditions on your device and shows partial sums separately.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": true,\n      \"inputMaxLength\": 1000,\n      \"inputMode\": \"text\",\n      \"label\": \"General term\",\n      \"placeholder\": \"3/2^n\",\n      \"help\": \"Use n as the index. Add a start such as n=2: before the term if needed. Up to 1,000 characters. Review the settings before running.\",\n      \"helpParts\": [\n        \"Use n as the index. Add a start such as n=2: before the term if needed.\",\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\nApply supported convergence tests after checking the original term domain at every included integer index. The report distinguishes a proved verdict from an inconclusive result; supported geometric series also get an exact sum.\n\n### How to use Series convergence checker\n\n1. Type the general term, such as 3/2^n, or set the start with a prefix such as n=2: 1/(n\\*ln(n)^2).\n2. Run the checker to apply the tests in order.\n3. Read the verdict, the deciding test and the preconditions of every test.\n4. Download the JSON report or the CSV files of tests and partial sums.\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- [Scientific calculator](https://happytails.ai/en/calculators/scientific-calculator/) ([Markdown](https://happytails.ai/en/calculators/scientific-calculator.md))\n\nTerms are limited to 1,000 characters and one index variable. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests.\n\n## Tests used\n\nThe checker applies the nth-term test, the geometric series test, the ratio and root tests, limit comparison with a p-series using the exact leading term c\\*n^p, direct comparison for bounded factors such as sin(n), the alternating series test with exact checks that the terms decrease to zero, and the integral test for 1/(n\\*ln(n)^p). Method: OpenStax, Calculus Volume 2, sections 5.2 to 5.6 (2016).\n\n### Preconditions\n\nOriginal denominators and other supported domain conditions are checked on the entire included integer tail before cancellation or zero-factor simplification. An included undefined term is rejected even when it occurs beyond the partial-sum table. When the full domain cannot be proved by the supported method, the checker returns a recoverable error. A test decides a verdict only when its preconditions are proved; otherwise the verdict remains inconclusive.\n\n### Partial sums\n\nPartial sums are listed for orientation. They are exact where the term evaluator returns exact rational values and otherwise use decimals, even when the terms are mathematically rational. They never prove convergence or divergence.\n"
}