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Generating functions are well-known to be much useful in combinatorics. But, maybe just since I am illiteral, all the applications coming in mind deal with power series, which are not just formal, but ...

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 from $k=1$ to $20$ ,The asymptotic of the titled series ...