Alternating Sum and Difference of Factorials to Infinity

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Theorem

According to Leonhard Paul Euler:

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



Proof


Sources