The figure under bracket represents divergent infinite series.
For n >100, the error % comes down to 0.1%.
0∫∞ xn e-x dx = n!
The Gamma function is represented as Γ(n)
Γ(n + 1) = nΓ(n)
Γ(n + 1) = n! and Γ(1) = 0! =1 and Γ(d/2 - 1)! = Γ(d/2)
Γ(1/2) = √Π
Γ(n )Γ(1-n) = Π / sin(Πn) where 0<n<1
From Stirling's formula it follows -
ln N! =NlnN -N + ln √ (2πn)
Neglecting the last term, ln N! =NlnN -N