Alternating Sum and Difference of Factorials to Infinity

 $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n n!$ $=$ $\ds \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u$ $\ds$ $=$ $\ds G$ the Euler-Gompertz constant $\ds$ $\approx$ $\ds 0 \cdotp 59634 \, 73623 \, 23194 \, 07434 \, 10784 \, 99369 \, 27937 \, 6074 \ldots$