# 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

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,59634 7355 \ldots$