{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(a_n)$ and $(b_n)$ be real sequences and let $L$ be a real number.\nAssume that $(b_n)$ is strictly increasing and that $b_n \\to +\\infty$ as $n \\to \\infty$.\nAssume that\n$$\\lim_{n \\to \\infty} \\frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L.$$\nThen $a_n / b_n \\to L$ as $n \\to \\infty$.\n\n## Remarks\n\nThere is also a $0/0$ form. If $a_n \\to 0$, $b_n \\to 0$, $(b_n)$ is strictly decreasing and the quotient of the differences tends to $L$, then $a_n / b_n \\to L$.\nWith $b_n = n$ it gives the [[convergence of the Cesàro means|0RDRPBMNAC]].\nFor example, with $a_n = 1 + 2 + \\cdots + n$ and $b_n = n^2$, the quotient of differences is $(n+1)/(2n+1) \\to 1/2$, so $a_n / n^2 \\to 1/2$.\nIt is a discrete analogue of [[L'Hôpital's rule|0JWT4PPQ7B]].\n\n## History\n\nOtto Stolz proved the $\\infty/\\infty$ case in a book of 1885. Ernesto Cesàro gave it in an article of 1888.\n\n## Explanations\n\n### L1 human\n\nTo compare two sequences that grow without bound, it is enough to compare their steps.\nIf each step of $a_n$ is about $L$ times the step of $b_n$, then $a_n$ is about $L$ times $b_n$ in the long run.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "40A05"
    ],
    "name": "Stolz–Cesàro theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1052752",
      "wikipedia": "Stolz–Cesàro theorem"
    },
    "sources": [
      {
        "locator": "Statement of the theorem for the */∞ case",
        "url": "https://en.wikipedia.org/wiki/Stolz–Cesàro_theorem"
      }
    ],
    "tag": "0Q3AKZV74D",
    "uses": [
      "0XKWSSY8P2",
      "0KN22AAW5R",
      "09AZCV6GMC"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0Q3AKZV74D/",
  "links": [
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    },
    {
      "field": "uses",
      "target": "0KN22AAW5R"
    },
    {
      "field": "uses",
      "target": "09AZCV6GMC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0Q3AKZV74D.md",
  "msc": [
    "40A05"
  ],
  "name": "Stolz–Cesàro theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0Q3/0Q3AKZV74D.md",
  "status": null,
  "tag": "0Q3AKZV74D",
  "uri": "m2://0Q3AKZV74D",
  "url": "https://math2.tamnd.com/wiki/stolz-cesaro-theorem/",
  "writing": {
    "tag": "0Q3AKZV74D",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}