The divergent sum of alternating factorials

I think, that an "analytical continuation" is impossible for this series because the powerseries in $x$ has zero-convergence radius and thus the method of recentering the series to extend its evaluatable range step-by-step cannot be exploited here.

Once I've found a nice method to sum this series and to assign a meaningful value to it (it's the same value found by Euler's method). I'm using a "matrix-method", but in some reverse order...
First I consider the infinite sized matrix of Eulerian numbers

    matrix E                  rowsums
  1   .   .   .  .  .  ...     1 = 0! 
  1   0   .   .  .  .          1 = 1!
  1   1   0   .  .  .          2 = 2!
  1   4   1   0  .  .          6 = 3!
  1  11  11   1  0  .         24 = 4!
  1  26  66  26  1  0   ...  120 = 5!
  ...  ...  ...  ... ...

To compute the rowsums in a matrix-formula let us introduce the column-vector of only ones $U=[1,1,1,1,...] ^\tau $ . The row-sums (the factorials per row) can then be expressed as $E \cdot U = F $ where $F=[0!,1!,2!,3!,...] ^\tau$ . Of course, if we rescale the rows by just that factorials we get $$ diag(F)^{-1} \cdot ( E \cdot U )= U $$ and if we have a series and write its terms in a rowvector $A=[a_0,a_1,a_2,...] $ we can express the summing of the series by $A \cdot U = s $ also by $$A \cdot \left( diag(F)^{-1} E U \right) $$ and even more, if the left dot-product has convergent summations we can even associate the matrix-product differently by $$ s = \left( (A \cdot diag(F)^{-1}) \cdot E \right) \cdot U $$ Now we see, that, given $A=[0!,-1!,2!,-3!,...]$ multiplied by the inverse factorials we have $ A \cdot diag(F)^{-1} = [1,-1,1,-1,1,-1,...] $ Let's denote this last vector by $W$, then the dot-products with the Eulerian numbers $W \cdot E$ give (columnwise) series expressions like the one with the first column: $$ w_0 =[1,-1,1,-1,1,-1,...] \cdot [1,1,1,1,1,...] = \sum_{k=0}^\infty (-1)^k = 1/2 $$ which is not a convergent series, but an alternating geometric series to which we assign the value of the closed fractional form $ {1 \over 1+1}=\frac 12$ .
With the following columns this is not so obvious. Fortunately we have an analytic description of that matrix-entries along the columns: they can be understood as sums of terms of geometric series and their derivatives again, for instance $ [0,1,4,11,26,...]=[1-1,2-2,2^2-3,2^3-4,2^4-5,2^5-6,...]$ and using this for the dotproduct with $W$ gives the sum for the alternating geometric series with quotient 2 minus the first derivative of the alternating geometric series qith quotient 1, which is explicitely $$ w_1 = W \cdot E_1 = \sum_{k=0}^\infty (-2)^k - \sum_{k=0}^\infty \left((-1)^k \cdot (k+1)\right) = \small {{1 \over 1+2} - \left(- \left({1 \over 1+1}\right)^2 + {1 \over 1+1}\right) = \frac 1{12}} $$ In a similar way we get for the next colum $w_2 = \frac 1{72}$ and so on. The final evaluation reads then $$ \left(W \cdot E \right) \cdot U = [{1\over2},{1\over 12},{1\over 72},{1\over 1080},...]\cdot U \approx 0.596347362323... $$ This is just an exemplaric illustration. Analytically it is better to introduce a continuous argument $x$ into the system, such that the vector $A$ becomes A(x)=$[1,1!x,2!x^2,3!x^3,...]$ and then evaluate that this holds for the indeterminate $x$ and then also for $A(x)_{|x=-1}=A(-1)$ by the identities for the alternating geometric series and their derivatives at $x$ .

A more complete explanation is this essay at my math-pages

From an engineering point of view, the concept is to apply to the "digital signal" represented by a sequence or sequence of partial sums, a Low-pass filtering so as to level-out the transient response however oscillating it might be, and even if with diverging amplitude.
What is retained is the $f = 0$ ( $f \to 0$) component.

So in the Grandi's series, with partial sums $1,0,1,0,\cdots$, the steady component = mean value is clearly $1/2$.
Also it is evident that in the Cesàro summation the average of the sequence of the partial sums is just the $a_0$ coefficient of the corresponding Fourier series, with $T \to \infty$ and thus with the Fourier series tending to the transform.

To focalize the steady component we need to compress the diverging (if it is) amplitude of the oscillation, by applying a filter with a stiffness of an order high enough and possibly having a mathematical formulation that pretty matches and annihilates the divergency of the "noise".

In conclusion a "filter" is conceptually the same as a sequence transformation and related summation approaches.
And same as the characteristics of the signal indicate the type and parameters of the filter which is better to adopt, so it is for the sequence transformation.