Answer
Diverges
Work Step by Step
Given: $a_n=\frac{n!}{2^{n}}$
$\lim\limits_{n \to \infty}\frac{n!}{2^{n}}=\lim\limits_{n \to \infty}\frac{1\times 2\times 3\times 4\times ....\times (n-1)\times n}{2\times 2 \times 2 \times 2\times 2\times ....\times2 \times 2}$
$=\lim\limits_{n \to \infty}\frac{1}{2}\times \frac{2}{2}\times \frac{3}{2} \times ....\times \frac{n-1}{2} \times \frac{n}{2}$
$=\lim\limits_{n \to \infty}\frac{1}{2}\times \frac{2}{2}\times [\frac{3}{2} \times ....\times \frac{n-1}{2} \times \frac{n}{2}]$
The bracket term part is a product of infinitely many terms, each of which is greater than $1.5$ after the first term.
Therefore, the bracket term part is greater than $1.5^{\infty}=\infty$
As we know $a^{\infty}=\infty$ if $a\gt 1$
Thus,
$=\lim\limits_{n \to \infty}\frac{1}{2}\times\frac{2}{2}\times [\infty ]$
$=\infty $
The sequence diverges.
