Let the sequence an be :

an = n!/n^n , and we know that : Lim n-------> infinity n!/n^n = 0 ,

also we know that n! =Γ(n+1),

Now an =Γ(x+1)/n^n ,

Now , is it possible to find the limit of this sequence when n⇒ infinity ,By using L'Hopital Rule by cosidering :

f(x) =Γ(x+1)/x^x ,

Lim x-------> infinity f(x) = ??? ( = 0 , by matlab)

Note : I did this limit by matlab and the limit was : 0 ,

I know that the derivative of the gamma function is :Γ(x)*psi(x) ,

How could we do this in L'Hopital Rule and simplifythat term to get zero finally ? ,

Best regards