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  "body": "\n## Summary\n\nBorel summation is a method that assigns a value to many [[divergent series|0NBRP6093V]], in particular to series whose coefficients grow like factorials.\nIt divides the $k$-th coefficient by $k!$, sums the resulting convergent series, and recovers the lost factorials with a [[Laplace transform|05XT5AVJBK]].\nThe method gives the usual sum for every convergent series.\nIt is a basic tool for the asymptotic series of perturbation theory and of differential equations.\n\n## Explanations\n\n### L1 human\n\nSome series, such as $1 - 1! + 2! - 3! + \\cdots$, grow too fast to have a sum in the usual sense.\nThe identity $k! = \\int_0^\\infty t^k e^{-t} \\, dt$ suggests a trick: replace $k!$ by this integral and exchange sum and integral.\nAfter dividing by $k!$, the series becomes a convergent one, and the integral then gives a finite number.\nFor this example the Borel sum is $\\int_0^\\infty e^{-t}/(1+t) \\, dt$, about $0.596$.\n",
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