{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(a_n)$ and $(b_n)$ be real sequences with $a_n \\ge 0$ and $b_n \u003e 0$ for all $n$.\nAssume that $a_n / b_n \\to c$ as $n \\to \\infty$, for a real number $c$ with $0 \u003c c \u003c \\infty$.\nThen $\\sum a_n$ converges if and only if $\\sum b_n$ converges.\n\n## Remarks\n\nFor large $n$ we have $\\frac{c}{2} b_n \\le a_n \\le 2c\\, b_n$. The [[comparison test|04Z5G0THN6]] then gives both directions.\nFor example, $\\frac{n}{n^3 + 1} / \\frac{1}{n^2} \\to 1$, so $\\sum n/(n^3 + 1)$ converges, like $\\sum 1/n^2$.\nIf $c = 0$, convergence of $\\sum b_n$ still implies convergence of $\\sum a_n$. The converse can fail.\n\n## Explanations\n\n### L1 human\n\nIf two series of positive terms have terms of the same size in the long run, they behave the same way.\nSo you can replace a complicated term by a simple one with the same leading behaviour.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "40A05"
    ],
    "name": "Limit comparison test",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6549441",
      "wikipedia": "Limit comparison test"
    },
    "sources": [
      {
        "locator": "Statement",
        "url": "https://en.wikipedia.org/wiki/Limit_comparison_test"
      }
    ],
    "tag": "0HW7BNHHGF",
    "uses": [
      "04Z5G0THN6",
      "0XKWSSY8P2"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HW7BNHHGF/",
  "links": [
    {
      "field": "uses",
      "target": "04Z5G0THN6"
    },
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HW7BNHHGF.md",
  "msc": [
    "40A05"
  ],
  "name": "Limit comparison test",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HW/0HW7BNHHGF.md",
  "status": null,
  "tag": "0HW7BNHHGF",
  "uri": "m2://0HW7BNHHGF",
  "url": "https://math2.tamnd.com/wiki/limit-comparison-test/",
  "writing": {
    "tag": "0HW7BNHHGF",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}