Yet another divergence joke:
$$\sum_{j=0}^\infty (-1)^j j!=e E_1(1)\approx 0.596347362\dots$$
where $E_n(z)$ is an exponential integral.
This is intimately related to the formal hypergeometric series ${}_2 F_0$ being the asymptotic expansion of a certain convergent improper integral.
This also serves as a warning of sorts to users of convergence acceleration methods like Wynn ε or Levin t ; attempts to sum a divergent series might give "correct" yet still unexpected answers.
