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Jaw-breaking sum (related to quasi-analytic decompositions of unity — and Hörmander’s Lemma 1.3.6 from LinPDE vol.1)
$$F\left(\frac 1{\text e}\right) ≈ 1-\frac 1{\text e};\qquad\text{here }F(z)≔\sum_{k=1}^\infty \frac{(k-1)^{k-1}}{k!}z^k.$$
The match is at least with 800 decimal places (checked with ...
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