Is $\log(n!)$ in $\Theta(n \log(n))$?

Question

I had two questions on my automated test which I don't understand the answer for.

$\log(n!) = \log(n\cdot (n-1)\cdot \cdots \cdot 2\cdot 1) = \log(n)+\log(n-1)+....+\log(1)$. So it is in $O(n\log(n))$. But is it also in $\Omega(n \log(n))$? I don't think so, but my automated interview test thought so!

$\log(n)+\log(n^2) = \log(n)+2\log(n) = 3\log(n)$. So, it is in $O(\log(n))$, $\Omega(\log(n))$ and $\Theta(\log(n))$. But for some reason my automated interview test thought otherwise.

Is my understanding correct or is the automated test correct?

Answer 1

You mentioned $$ \log(n!) = \log(n(n-1)\cdots1) = \log(n)+\log(n-1)+ \cdots +\log(1) $$ From this, we can write (assuming $\log$ is base 2) $$ \begin{array}{l} \log(n!) \\ = \log(n)+\log(n-1)+....+\log(1) \\ \geq \log(n)+ \cdots + \log(n/2) \\ \geq \log(n/2)+ \cdots + \log(n/2) \\ = n/2 \cdot \log(n/2) \\ = n/2 \cdot (\log(n) - 1) \\ = n/2 \cdot \log(n) - n/2 \\ \end{array} $$ since $\lim_{n\rightarrow +\infty} \dfrac{n/2}{n/2 \cdot \log(n)} = 0$, the last expression above is $\Theta(n \log(n))$, hence $\log(n!)$ is $\Omega(n \log(n))$.

Answer 2

Yes. This follows from Stirling's approximation: $\sqrt{2\pi}\,n^{n+1/2}\,e^{-n}\leq n!\leq e\,n^{n+1/2}\,e^{-n}$.