# 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 | |||||||||||

\(\ds \) | \(\approx\) | \(\ds 0 \cdotp 59634 \, 73623 \, 23194 \, 07434 \, 10784 \, 99369 \, 27937 \, 6074 \ldots\) |

This article, or a section of it, needs explaining.In particular: Clarify meaning of this equality. Using naive manipulations (carelessly swapping integration/sum and using geometric series results) it's quite straightforward to see why someone might think this is true but we should establish the sense in which it is, since the sum on the LHS diverges.What we need to do is go back to Euler's original statement of this and see what he meant. He did lots of stuff like this, plugging values into formulae that they weren't applicable to, like e.g. $1 + 2 + 3 + \ldots = \dfrac 1 {12}$, I presume it's like one of those. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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