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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 [[smooth|0BE3QDH73A]].\nThe Taylor series of $f$ at $a$ is the [[power series|0VP56B4EMX]]\n$$\\sum_{n=0}^{\\infty} \\frac{f^{(n)}(a)}{n!} (x - a)^n.$$\n\n## Remarks\n\nIts partial sums are the [[Taylor polynomials|0HFSFVMWQB]] of $f$ at $a$.\nThe Taylor series may diverge for each $x \\ne a$. It may also converge to a function other than $f$, as for the [[smooth non-analytic function|09RTYYHXKA]].\nThe Taylor series converges to $f$ near $a$ exactly when $f$ is [[analytic|0JD5ZNW0F4]] at $a$.\nFor $a = 0$ it is the [[Maclaurin series|043SJ5RF59]].\n\n## Explanations\n\n### L1 human\n\nThe Taylor series is the Taylor polynomial of infinite degree.\nIt uses all the derivatives of $f$ at one point.\n",
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