Alternating Sum and Difference of Factorials to Infinity

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Theorem

According to Leonhard Paul Euler:

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



Proof


Sources