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  "body": "\n## Summary\n\nAn absolutely monotonic function on an interval is a [[smooth function|0V33BNKZ9S]] whose derivatives of every order, including order $0$, are nonnegative there.\nThe mirror notion is a completely monotone function, whose derivatives alternate in sign.\nOn the half-lines $(-\\infty, 0)$ and $(0, \\infty)$ respectively, both classes are described by [[Bernstein's theorem on monotone functions|ZBS1JA62QP]] as mixtures of exponential functions.\n\n## Explanations\n\n### L1 human\n\nThe function is nonnegative, increasing, convex, and so on for every higher derivative.\nThe exponential function $e^x$ is the main example: all its derivatives equal $e^x \u003e 0$.\nA polynomial with nonnegative coefficients is absolutely monotonic on $[0, \\infty)$.\nA power series with nonnegative coefficients is absolutely monotonic on $[0, r)$, where $r$ is its radius of convergence.\n",
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