calculus - What is the limit of the sequence n!/4^n?

I am trying to find the limit of the sequence by using root test and I don't understand why the limit is not zero?

(the answer is inf).

Answer

By the root test:

$$\begin{array}{rcl}
\displaystyle \limsup_{n\to\infty} \sqrt[n]{a_n} &=& \displaystyle \limsup_{n\to\infty} \sqrt[n]{\dfrac{n!}{4^n}} \\
&=& \displaystyle \dfrac14 \limsup_{n\to\infty} \sqrt[n]{n!} \\
&=& \displaystyle \dfrac14 \limsup_{n\to\infty} \sqrt[n]{\exp\left(n \ln n - n\right)} \\
&=& \displaystyle \dfrac14 \limsup_{n\to\infty} \exp\left(\ln n - 1\right) \\
&=& \displaystyle \dfrac1{4e} \limsup_{n\to\infty} n \\

&=& \infty
\end{array}$$

Hence the sequence diverges to infinity.