<s>[INST] Logic behind the multiplicative form of "Gauss trick"? I was going through $ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
I don't follow why/how this is true $(n!)^2 = (1 \cdot n) (2 \cdot (n-1)) (3 \cdot (n-2)) \cdots ((n-2) \cdot 3) ((n-1) \cdot 2) (n \cdot 1)\ge n^n$
I read that At least half the terms in the product for n! will be at least n/2 .. so ?
Please advise.
 [/INST] If $k \le \frac{n-1}2$, then $4k^2 \le n^2 - 2n + 1$, and so
\begin{align}
\left(\frac{n + 1}2 + k\right)\left(\frac{n + 1}2 - k\right)
& = \left(\frac{n+1}2\right)^2 - k^2 \\
& = \frac{n^2 + 2n + 1 - 4k^2}{4} \\
& \ge \frac{n^2 + 2n + 1 - (n^2 - 2n + 1)}{4} \\
& = n.
\end{align}
</s>