Answer
The series diverges
Work Step by Step
$if\lim _{n\rightarrow \infty }a_{n}\neq 0\Rightarrow \sum ^{n}_{1}a_{n}$ diverges
$\lim _{n\rightarrow \infty }\dfrac {n!}{2^{n}}=\dfrac {1\times 2\times 3\times \ldots \times n}{2\times 2\times 2\ldots \times 2}=\dfrac {1}{2}\times \dfrac {2}{2}\times \dfrac {3}{2}\times \ldots \dfrac {n}{2}=\infty \neq 0$
So the series diverges