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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 0}$ be any sequence of real numbers.\nThen there is a [[smooth function|0BE3QDH73A]] $f \\colon \\R \\to \\R$ such that $f^{(n)}(0) = a_n$ for every $n \\ge 0$.\n\n## Remarks\n\nIdea of the proof: choose a [[bump function|HS65DW0HXH]] $\\chi$ equal to $1$ on $[-1/2, 1/2]$ and $0$ outside $[-1, 1]$.\nSet $f(x) = \\sum_{n \\ge 0} \\frac{a_n}{n!} x^n \\chi(\\lambda_n x)$ with numbers $\\lambda_n \\ge 1$ that grow fast.\nThe $n$-th term agrees with $\\frac{a_n}{n!} x^n$ near $0$.\nFor $k \u003c n$, its $k$-th derivative has supremum at most $2^{-n}$ when $\\lambda_n$ is large enough.\nThen every series of derivatives converges uniformly, so [[term-by-term differentiation|0CS42KEM5C]] gives $f^{(n)}(0) = a_n$.\n\nThe construction also gives $f$ with compact support.\nExample: for $a_n = (n!)^2$, the [[Taylor series|0225QPWRRE]] of $f$ at $0$ is $\\sum n! \\, x^n$, which converges only at $x = 0$.\n\n## Explanations\n\n### L1 human\n\nAny list of numbers can be the list of derivatives at $0$ of some smooth function.\nThere is no growth condition, unlike for analytic functions, whose derivatives at a point can grow at most like $C R^n n!$.\nSo every formal power series is the Taylor series of some smooth function.\n\n### L2 human\n\nThe lemma says that the map sending a smooth function to its Taylor series at $0$, a [[formal power series|072W67KFYK]], is onto.\nIt is far from one-to-one: the [[smooth non-analytic function|09RTYYHXKA]] equal to $e^{-1/x}$ for $x \u003e 0$ and to $0$ for $x \\le 0$ has zero Taylor series at $0$.\nTogether these facts show how loosely the Taylor series controls a smooth function.\nThe lemma has versions in several variables and with parameters, and it is used in the study of singularities and of partial differential equations.\n",
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