January 3rd, 2017, 07:28 AM  #1 
Newbie Joined: Jan 2017 From: Nowhere Posts: 5 Thanks: 1  Need help with (n^n)/((n!)^2)
I m struggling with this limit for quite a while.Can someone help me please? $\displaystyle (n^n)/(n!)^2 $ as n approaches infinity 
January 3rd, 2017, 01:48 PM  #2 
Member Joined: Apr 2015 From: USA Posts: 46 Thanks: 32 
Try Stirling's approximation for factorials: $\displaystyle n!\approx n^n e^{n}\sqrt{2\pi n}$ Then you can deal with $\displaystyle\lim_{n\to\infty}\frac{n^n}{\left(n^ n e^{n}\sqrt{2\pi n}\right)^2}$ 
January 3rd, 2017, 03:10 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,660 Thanks: 965 Math Focus: Elementary mathematics and beyond 
Another approach: $\displaystyle \frac{n\cdot n\cdot n\cdots}{n^2\cdot(n1)^2\cdot(n2)^2\cdots}$ The terms in the denominator are greater than the terms in the numerator until $na\leq\sqrt n$ for some $a$. But what is the ratio of a root to its square? It is $\frac{1}{\sqrt n}$ and we have $$\lim_{n\to\infty}\frac{1}{\sqrt n}=0$$ Hence we may state $$\lim_{n\to\infty}\frac{n^n}{(n!)^2}=0$$ 
January 5th, 2017, 10:16 AM  #4 
Newbie Joined: Jan 2017 From: Nowhere Posts: 5 Thanks: 1 
Thank you very much for your answers!
