x! - (x - 1)! = (x - 1) (x - 1)!
=
(x-1)^2*(x-2)!
=
x^3 (x - 3)! - 4 x^2 (x - 3)! + 5 x (x - 3)! - 2 (x - 3)!
=
etc .... en sortant x-y de (x-y)!
x! - (x - 1)! = (x - 1)^2 (x - 10)! (x - 2) (x - 3) (x - 4) (x - 5) (x - 6) (x - 7) (x - 8) (x - 9)
x! - (x - 1)! = x^10 (x - 10)! - 46 x^9 (x - 10)! + 915 x^8 (x - 10)! - 10320 x^7 (x - 10)! + 72723 x^6 (x - 10)! - 332598 x^5 (x - 10)! + 993005 x^4 (x - 10)! - 1896380 x^3 (x - 10)! + 2199276 x^2 (x - 10)! - 1389456 x (x - 10)! + 362880 (x - 10)!
x! - (x - 1)! = (x - 1)^2 (x^8 - 44 x^7 + 826 x^6 - 8624 x^5 + 54649 x^4 - 214676 x^3 + 509004 x^2 - 663696 x + 362880) (x - 10)!
Factorielle décroissante ...............
![\[\boxed{\sum_{k=1}^{n}k\thinspace k!=\left(n+1\right)!-1}\]](https://math-os.com/wp-content/ql-cache/quicklatex.com-1270d90d740efc7d11dd0892673cb7ae_l3.svg "Rendered by QuickLaTeX.com")
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