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Summation of divergent series is applied in dynamical systems, q-difference equations, and mathematical physics nowadays (for example, see that book for info), this is not an old-fashioned topic.

Mathematica includes, in particular, Borel summation. More exactly, the Regularization->Borel option of the Sum command is the implementation of the Borel's integral summation method with analytic continuation in Mathematica.

Trying Regularization->Borel in version 13 on Windows 10, I face a problem with the result of

Sum[n!, {n, 0, Infinity}, Regularization -> "Borel"]

-Subfactorial[-1]

N[-Subfactorial[-1]]

0.697175 + 1.15573 I

But directly using the definition of Regularization -> "Borel" in the "Details" section of the documentation to Regularization, I obtain

Sum[n!*t^n/n!, {n, 0, Infinity}]

1/(1-t)

in a neighborhood of the origin.

First, the function f[t_]:=1/(1-t) cannot be analytically continued along the positive ray because of its singularity at t==1 (e.g. see Encyclopedia of Mathematics and Wiki for info).

Second, leaving aside the analytical continuation, the integral $\int_0^\infty \frac {e^{-t}} {1-t} \, dt$ diverges ant its principal value

Integrate[1/(1 - t)*Exp[-t], {t, 0, Infinity},  PrincipalValue -> True] // N

0.697175

This numerically coincides with the real part of the Mathematica result. It should be stressed that Wiki and Encyclopedia of Mathematics and G. H. Hardy's "Divergent series" say nothing about using principal values of integrals in Borel summation.

The article Borel summation of Wiki presents an example in the "An alternating factorial series" section $$ \sum _{k=0}^\infty k!(-z)^k =\frac {e^{\frac 1 z}} z \Gamma \left(0,\frac 1 z\right ), $$

but Wiki clearly says that is valid only in the so-called Borel polygon $\Re z > 0$ and Encyclopedia of Mathematics seconds it. The Google search $\sum_{k=0}^\infty k!$ does not bring the Mathematica result for it to me.

How to explain and ground the Mathematica result?

Edit. A typo: $\Re z > 0$ instead of $\Re z > 1$ .

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My explanation, but not a justification, of the Mathematica result is as follows.

Sum[n!*(-z)^n, {n, 0, Infinity}, Regularization -> "Borel"] /. z -> -1

-(Gamma[0, -1]/E)

N[%]

0.697175 + 1.15573 I

The substitution z->-1 in the above is wrong because $-1$ does not belong to the Borel polygon as it is explained in Wiki and G. H. Hardy's "Divergent series". If we allow such values, then (as far as I understand it) such a modification of Borel summation does not possess usual properties of a regularization: regularity, linearity, and stability. The correct result should be the returned input (as in many similar cases, say Sum[n!/(n + 1), {n, 0, Infinity}, Regularization -> "Borel"]).

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