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  "body": "\n## Summary\n\nA completely monotone function is an infinitely differentiable function on $(0, \\infty)$ whose derivatives alternate in sign: $f \\ge 0$, $f' \\le 0$, $f'' \\ge 0$, and so on.\nIt is the mirror image of an [[absolutely monotonic function|QQSPWH4NG9]] under $x \\mapsto -x$.\nBy [[Bernstein's theorem|ZBS1JA62QP]], the completely monotone functions are exactly the [[Laplace transforms|0V49JEZW06]] of positive measures on $[0, \\infty)$.\nThis makes them central in probability and potential theory.\n\n## Explanations\n\n### L1 human\n\nA completely monotone function is positive, decreasing, convex, and every further derivative keeps this alternating pattern.\nThe function $e^{-x}$ is the basic example: its derivatives are $\\pm e^{-x}$ with alternating signs.\nAny sum of decaying exponentials $c_1 e^{-t_1 x} + c_2 e^{-t_2 x}$ with $c_i \\ge 0$, $t_i \\ge 0$ is again completely monotone.\nSo is $1/x$, which is a continuous mixture of such exponentials.\n",
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