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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open [[interval|0Q1D85B8G6]] and let $f \\colon I \\to \\R$ be [[infinitely differentiable|0V33BNKZ9S]].\nThe function $f$ is *absolutely monotonic* on $I$ if $f^{(k)}(x) \\ge 0$ for every integer $k \\ge 0$ and every $x \\in I$.\nHere $f^{(0)} = f$ and $f^{(k)}$ is the [[derivative of order $k$|04TKTF8KDK]].\nThe function $f$ is *completely monotone* on $I$ if $(-1)^k f^{(k)}(x) \\ge 0$ for every $k \\ge 0$ and every $x \\in I$.\n\n## Remarks\n\nExample: $f(x) = e^{x}$ on $\\R$ is absolutely monotonic, since $f^{(k)}(x) = e^x \u003e 0$ for all $k$.\nLikewise $g(x) = e^{-x}$ is completely monotone, since $(-1)^k g^{(k)}(x) = e^{-x} \u003e 0$.\nNon-example: $f(x) = x^3$ on $(-1, 1)$ fails at $k = 0$, since $f(-\\tfrac{1}{2}) \u003c 0$, although $f^{(3)} = 6 \u003e 0$.\nThe function $f$ is absolutely monotonic on $I$ if and only if $x \\mapsto f(-x)$ is completely monotone on $-I$.\nOn an interval that contains an endpoint, such as $[0, \\infty)$, one asks the inequalities on the interior and continuity of $f$ at the endpoint.\n\n## Explanations\n\n### L1 human\n\nEvery derivative, starting with the function itself, is nonnegative.\nSo the function is nonnegative, increasing, convex, and its convexity keeps increasing, without end.\n",
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