{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $k \\ge 1$, let $I$ be an open interval, let $a \\in I$, and let $f \\colon I \\to \\mathbb{R}$ be $k$ times differentiable at $a$.\nThen\n$$f(x) = \\sum_{j=0}^{k} \\frac{f^{(j)}(a)}{j!} (x - a)^j + o\\big((x - a)^k\\big) \\quad \\text{as } x \\to a.$$\n\n## Remarks\n\nThis is [[Taylor's theorem|03QHBVW4TT]] written with [[little o notation|04TSHB9H8D]].\nFor example, $\\cos x = 1 - x^2/2 + o(x^3)$ as $x \\to 0$.\nMathlib states a form for $C^k$ functions on a convex set as `taylor_isLittleO`.\n\n## Explanations\n\n### L1 human\n\nThe Peano form only says that the error is small compared with $(x - a)^k$.\nIt gives no bound for a fixed $x$, but it needs weak hypotheses.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "41A58"
    ],
    "name": "Taylor's theorem with the Peano remainder",
    "role": "corollary",
    "sources": [
      {
        "locator": "Statement of the theorem",
        "url": "https://en.wikipedia.org/wiki/Taylor%27s_theorem"
      }
    ],
    "tag": "03AJHX431A",
    "uses": [
      "03QHBVW4TT",
      "04TSHB9H8D"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/03AJHX431A/",
  "links": [
    {
      "field": "uses",
      "target": "03QHBVW4TT"
    },
    {
      "field": "uses",
      "target": "04TSHB9H8D"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/03AJHX431A.md",
  "msc": [
    "41A58"
  ],
  "name": "Taylor's theorem with the Peano remainder",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/03A/03AJHX431A.md",
  "status": null,
  "tag": "03AJHX431A",
  "uri": "m2://03AJHX431A",
  "url": "https://math2.tamnd.com/wiki/taylors-theorem-with-the-peano-remainder/",
  "writing": {
    "tag": "03AJHX431A",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}