Stirling's Formula

where 0<θ<1.

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