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  "body": "\n## Definition\n\nLet $\\sum_{k \\ge 0} a_k$ be a [[series|0JH4MN4K1C]] of complex numbers.\nIts *Borel transform* is the [[power series|0VP56B4EMX]]\n$$\\mathcal{B}(t) = \\sum_{k \\ge 0} \\frac{a_k}{k!} t^k.$$\nThe series $\\sum_k a_k$ is *Borel summable* if:\n\n1. $\\mathcal{B}$ has a positive [[radius of convergence|0660QX1VPR]];\n2. $\\mathcal{B}$ has an [[analytic continuation|0M3YQJ81NB]] to an open set that contains the half-line $[0, \\infty)$, still written $\\mathcal{B}$;\n3. the improper integral $\\int_0^\\infty e^{-t} \\mathcal{B}(t) \\, dt$ converges.\n\nThe value of this integral is the *Borel sum* of the series.\n\n## Remarks\n\nExample: for $a_k = z^k$ with a complex number $z$, $\\mathcal{B}(t) = e^{zt}$, and the integral converges exactly when $\\operatorname{Re} z \u003c 1$, with value $1/(1 - z)$.\nSo the geometric series is Borel summable to $1/(1-z)$ on the half-plane $\\operatorname{Re} z \u003c 1$, also where it diverges, such as at $z = -1$, where the sum is $1/2$.\nNon-example: for $a_k = (k!)^2$, the Borel transform $\\sum_k k! \\, t^k$ has radius of convergence $0$, so clause 1 fails.\nThe method is regular: a convergent series is Borel summable, and its Borel sum is its ordinary sum.\nFor a power series $\\sum_k a_k z^k$, the Borel sum is $\\int_0^\\infty e^{-t} \\mathcal{B}(zt) \\, dt$, which is a [[Laplace transform|05XT5AVJBK]] of $\\mathcal{B}$ up to a change of variable.\nA weaker variant, the weak Borel method, uses the limit of $e^{-t} \\sum_{n \\ge 0} s_n t^n/n!$ as $t \\to \\infty$, where $s_n$ are the partial sums.\n",
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