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  "body": "\n## Statement\n\nLet $k \\ge 0$ and let $a \\ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.\nLet $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.\nThen there is $\\xi$ strictly between $a$ and $x$ with\n$$f(x) = P_k(x) + \\frac{f^{(k+1)}(\\xi)}{k!} (x - \\xi)^k (x - a),$$\nwhere $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.\n\n## Remarks\n\nApply the [[mean value theorem|0BARQY52N5]] to $F(t) = f(t) + \\sum_{j=1}^{k} f^{(j)}(t) (x - t)^j / j!$ on $J$. Note that $F(x) = f(x)$ and $F(a) = P_k(x)$.\nThis form sometimes gives better bounds than the [[Lagrange form|0H5ZAZCX2T]], for example for $\\ln(1 + x)$ with $-1 \u003c x \u003c 0$.\nMathlib states it as `taylor_mean_remainder_cauchy`.\n\n## Explanations\n\n### L1 human\n\nThe Cauchy form is another exact expression for the error.\nIt moves part of the power of $(x - a)$ to a power of $(x - \\xi)$.\n",
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  "msc": [
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  "name": "Taylor's theorem with the Cauchy remainder",
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