A professor in class gave the following lower bound for the factorial $$ n! \ge {\left(\frac n2\right)}^{\frac n2} $$ but I don't know how he came up with this formula. The upper bound of $n^n$ was quite easy to understand. It makes sense. Can anyone explain why the formula above is the lower bound?
Any help is appreciated.
Suppose first that $n$ is even, say $n=2m$. Then
$$n!=\underbrace{(2m)(2m-1)\ldots(m+1)}_{m\text{ factors}}m!\ge(2m)(2m-1)\ldots(m+1)>m^m=\left(\frac{n}2\right)^{n/2}\;.$$
Now suppose that $n=2m+1$. Then
$$n!=\underbrace{(2m+1)(2m)\ldots(m+1)}_{m+1\text{ factors}}m!\ge(m+1)^{m+1}>\left(\frac{n}2\right)^{n/2}\;.$$
Hint: For a positive integers $a$, we have $a(n-a)\ge \frac{n}{2}$. This is because for $0\le x\le n$, the function $x(n-x)$ is increasing up to $x=\frac{n}{2}$, and then decreasing.
Pair the numbers $a$ and $n-a$ in the factorial.
As an alternative to the very fine methods suggested, we can use that
$$n!\ge \frac{n^n}{e^{n-1}}$$
which can easily proved by induction and then show that
$$\frac{n^n}{e^{n-1}}\ge {\left(\frac n2\right)}^{\frac n2} \iff 2n\ge e^{2-\frac2n}$$
which is true for any $n$ (indeed for $n\ge 4 \implies 2n\ge 8 \ge e^2$ and by inspection for $n=1,2, 3$).
