sum_(n=0)^∞ (n! t^n)/(n!) = 1/(1 - t)

n! is a sequence with integer values for nonnegative n.

n | n! 0 | 1 1 | 1 2 | 2 3 | 6 4 | 24 5 | 120 6 | 720 7 | 5040 8 | 40320 9 | 362880 10 | 3628800

Γ(n + 1)

(no roots exist)

1 - gamma n + 1/12 (6 gamma ^2 + π^2) n^2 + 1/6 n^3 (- gamma ^3 - ( gamma π^2)/2 + polygamma(2, 1)) + 1/24 n^4 ( gamma ^4 + gamma ^2 π^2 + (3 π^4)/20 - 4 gamma polygamma(2, 1)) + O(n^5) (Taylor series)

e^(-n) n^n (sqrt(2 π) sqrt(n) + 1/6 sqrt(π/2) sqrt(1/n) + 1/144 sqrt(π/2) (1/n)^(3/2) - (139 sqrt(π/2) (1/n)^(5/2))/25920 - (571 sqrt(π/2) (1/n)^(7/2))/1244160 + (163879 sqrt(π/2) (1/n)^(9/2))/104509440 + O((1/n)^(11/2)))

d/dn(n!) = Γ(n + 1) polygamma(0, n + 1)

n! = Γ(1 + n)

n! = (1)_n

n! = (-1 + n)!! n!!

n! = sum_(k=0)^∞ ((n - n_0)^k Γ^(k)(1 + n_0))/(k!) for ((n_0 not element Z or n_0>=0) and n->n_0)

n!∝exp(-n + sum_(k=0)^∞ (n^(-1 - 2 k) B_(2 + 2 k))/(2 + 6 k + 4 k^2)) n^(1/2 + n) sqrt(2 π) for (abs(arg(n))<π and abs(n)->∞)

n!∝e^(-n) n^(1/2 + n) sqrt(2 π) + e^(-n) n^(1/2 + n) sqrt(2 π) sum_(k=1)^∞ sum_(j=1)^(2 k) ((-1)^j 2^(-j - k) n^(-k) D_(2 (j + k), j))/((j + k)!) for ((abs(arg(n))<π and abs(n)->∞ and D_(m, j) = (-1 + m) ((-2 + m) D_(-3 + m, -1 + j) + D_(-1 + m, j)) and D_(0, 0) = 1 and D_(m, 1) = (-1 + m)! and D_(m, j) = 0) for m<=3 j - 1)

n! = integral_0^1 log^n(1/t) dt for Re(n)>-1

n! = integral_0^∞ e^(-t) t^n dt for (n element Z and n>=0)

n! = integral_1^∞ e^(-t) t^n dt + sum_(k=0)^∞ (-1)^k/((1 + k + n) k!)

n! = integral_0^∞ t^n (e^(-t) - sum_(k=0)^m (-t)^k/(k!)) dt for (m element Z and m>0 and -1