Answer
converges to $4$
Work Step by Step
As we know that when $x \gt 0$ so, $\lim\limits_{n \to \infty} \sqrt[n] {n}=1$ and $\lim\limits_{n \to \infty} x^{1/n}=1$
Let $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \sqrt [n] {4^n n}$
This implies that $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \sqrt [n] {4^n n}$
and $a_n=(4) \lim\limits_{n \to \infty}\sqrt [n] {n} =4 \cdot 1=4$
Thus, $\lim\limits_{n \to \infty} a_n=4$ and {$a_n$} converges to $4$