Formal way to proof that $k! > (k/2)^{(k/2)}$ for any $k \in \{1,2,3,4...\}$

Question

I am looking for a formal way to proof that $k! > (k/2)^{(k/2)}$ for any $k \in \{1,2,3,4...\}$

I tried proof by induction but this didn't really work out. So I kind of know why $k! > (k/2)^{(k/2)}$, but can't come up with a good way to write it down.

Answer 1

Case 1: $k$ is odd. Then $\frac{k+1}{2},\frac{k+3}{2},...,\frac{k+k}{2}$ are all greater than $\frac k2$ and are factors in $k!$. How many of these factors are there?

You can do case 2 on your own.

Answer 2

$k! = k(k-1)(k-2)\ldots2.1 \geq k(k-1)\ldots \lceil \frac{k}{2} \rceil \geq \left(\lceil \frac{k}{2} \rceil \right) \left(\lceil \frac{k}{2} \rceil \right) \ldots \left(\lceil \frac{k}{2} \rceil \right) = \lceil \frac{k}{2} \rceil^{\lceil \frac{k}{2} \rceil} \geq \left(\frac{k}{2}\right)^{\frac{k}{2}}$

Answer 3

I'll give you a simple example... Consider n=100... You have 100=100.1, 100<99.2, 100<98.3,... 100<51.50... Multiplying this inequalities you obtain 50^50<100^50<100!