<s>[INST] Show that $e^n>\frac{(n+1)^n}{n!}$ without using induction. I have got an inequality problem which is as follow:

Show that $e^n>\frac{(n+1)^n}{n!}$

I can do it by induction but I have been told to prove it without induction.
My Work:
$$e^n=1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+........$$
$$e^n>1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+........+\frac{n^n}{n!}$$
$$e^n>\frac{n^n}{n!}+\frac{n^{n-1}}{(n-1)!}.......+\frac{n^2}{2!}+n+1$$
From here I can't go further. 
I shall be thankful if you guys can provide me a complete solution/proof of this inequality. A hint will also work.
Thanks in advance.
 [/INST] $$e^n=1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+........$$
$$e^n>1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+........+\frac{n^n}{n!}$$
$$e^n>\frac{n^n}{n!}+\frac{n^{n-1}}{(n-1)!}.......+\frac{n^2}{2!}+n+1$$ $$e^n>n^n\left[\frac{1}{n!}+\frac{1}{n(n-1)!}+\frac{1}{n^2(n-2)!}...+\frac{1}{n^{n-1}}+\frac{1}{n^n}\right] $$ $$e^n>\frac{n^n}{n!}\left[1+\frac{1}{n}n+\frac{1}{n^2}n(n-1)+\frac{1}{n^3}n(n-1)(n-2)...+\frac{n!}{n^n}\right] $$  $\because$ $$n(n-1)>\frac{n(n-1)}{2!}$$and $$n(n-1)(n-2)>\frac{n(n-1)(n-2)}{3!}$$and $$n!>1$$ 
$\therefore $ $$e^n>\frac{n^n}{n!}\left[1+n\frac{1}{n}+\frac{n(n-1)}{2!}\frac{1}{n^2}+...+\frac{1}{n^n}\right]$$ $$e^n>\frac{n^n}{n!}(1+\frac1n)^n$$ $$e^n>\frac{n^n}{n!}\frac{(n+1)^n}{n^n}$$  $$e^n>\frac{(n+1)^n}{n!}$$
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