let $ S = \sum\_{k=0}^\infty (-1)^k (k!)a\_k $ a divergent series such that $b\_k=(-1)^k (k!)a\_k >0 $ for $k>1$ , and $b\_k$ signed [this](https://oeis.org/A214645/list) from $k=1$ to $20$ ,The asymptotic of the titled series is :$$ A(x) = x + x^2/2! + 3\*x^3/3! + 16\*x^4/4! + 126\*x^5/5! + 1333\*x^6/6!+\cdots$$



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> > **Question:**
> >  Is this : $ S = \sum\_{k=0}^\infty (-1)^k (k!)a\_k $ a Borel summable with $ a\_k$ altern sequence?
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Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?
**Note**:This is not an answer but probably helping you to get the answer of your question.just to show some properties related to your formel series .
we denote by $f$ your divergent power series
,The formel series you have is non analytic function and it is smooth ,it's has a positive radius of convergence.Then, it extends uniquely by analytic continuation to a maximally-defined analytic function , Note that the Taylor series of $f$ at $0$ has non-negative coefficients, and it's seems the coefficients are not grow fast that is one of the weaker condition to judge that the formel series you have can be Borel summable


