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  "body": "\n## Example\n\nThe alternating series of factorials is\n$$\\sum_{k=0}^{\\infty} (-1)^k k! = 1 - 1 + 2 - 6 + 24 - 120 + \\cdots.$$\nIts terms do not tend to $0$, since $k! \\to \\infty$.\nSo it is a [[divergent series|0JWMJSTBSG]].\n\n## Remarks\n\nThe series is [[Borel summable|RQYX6E42D0]].\nIts Borel transform is $\\sum_{k \\ge 0} \\frac{(-1)^k k!}{k!} t^k = \\sum_{k \\ge 0} (-t)^k = \\frac{1}{1 + t}$ for $\\abs{t} \u003c 1$.\nThe function $1/(1+t)$ is analytic on a neighbourhood of $[0, \\infty)$, and the Borel sum is\n$$\\int_0^\\infty \\frac{e^{-t}}{1 + t} \\, dt \\approx 0.596347.$$\nThis number is known as the Gompertz constant.\nEuler studied this series and gave it the same value.\nThe formal series $\\sum_k (-1)^k k! \\, x^k$ is the asymptotic expansion of $\\int_0^\\infty \\frac{e^{-t}}{1 + x t} \\, dt$ as $x \\to 0^+$, and the example is this series taken formally at $x = 1$.\n\n## Explanations\n\n### L1 human\n\nThe terms of this series grow so fast that the partial sums swing wildly between large positive and large negative numbers.\nStill, one can give it a meaningful value.\nDivide each term by $k!$ to get the geometric series of $1/(1+t)$, then undo the division with the integral $\\int_0^\\infty e^{-t} t^k \\, dt = k!$.\nThis gives about $0.5963$.\n",
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