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  "body": "\n## Definition\n\nLet $I$ be an open interval, let $a \\in I$, and let $f \\colon I \\to \\mathbb{R}$ be $k$ times differentiable at $a$.\nThe Taylor polynomial of degree $k$ of $f$ at $a$ is\n$$P_k(x) = \\sum_{j=0}^{k} \\frac{f^{(j)}(a)}{j!} (x - a)^j = f(a) + f'(a)(x - a) + \\cdots + \\frac{f^{(k)}(a)}{k!} (x - a)^k.$$\n\n## Remarks\n\nIt is the unique polynomial $P$ of degree at most $k$ with $P^{(j)}(a) = f^{(j)}(a)$ for $j = 0, \\dots, k$, using [[higher derivatives|084NGSTWM7]].\nFor $k = 1$, its graph is the [[tangent line|01R1HW3Y42]].\nThe difference $R_k = f - P_k$ is the remainder. [[Taylor's theorem|03QHBVW4TT]] says that it is small near $a$.\nMathlib has `taylorWithinEval` for Taylor polynomials within a set.\n\n## Explanations\n\n### L1 human\n\nThe Taylor polynomial copies the value and the first $k$ derivatives of $f$ at $a$.\nIt is the polynomial that best fits $f$ near $a$.\n",
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