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  "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 $a$.\nLet $P_k$ be the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.\nThen there is a function $h_k \\colon I \\to \\mathbb{R}$ with $h_k(x) \\to 0$ as $x \\to a$ and\n$$f(x) = P_k(x) + h_k(x) (x - a)^k \\quad \\text{for all } x \\in I.$$\n\n## Remarks\n\nIn [[little o notation|04TSHB9H8D]], the conclusion is the [[Peano form|03AJHX431A]] $f(x) - P_k(x) = o((x - a)^k)$.\nUnder stronger hypotheses, the remainder has explicit forms: the [[Lagrange form|0H5ZAZCX2T]], the [[Cauchy form|005Q84BMAD]], and the [[integral form|0WJGR81HBD]].\nFor $k = 1$, it says that $f(x) = f(a) + f'(a)(x - a) + o(x - a)$.\n\n## Explanations\n\n### L1 human\n\nNear $a$, a function is close to its Taylor polynomial.\nThe error is small even compared with $(x - a)^k$.\n",
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