Answer
converges to $0$.
Work Step by Step
As we know that a sequence converges when $\lim\limits_{n \to \infty}a_n$ exists.
Consider $a_n=\dfrac{\ln (n^2)}{n}$
Re-write the given sequence as:$a_n=\dfrac{2 \ln n}{n}$
Apply limits to both sides.
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\dfrac{2 \ln n}{n}$
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\frac{\infty}{\infty}$
Since, we can see that the limit has the form of $\frac{\infty}{\infty}$, use L-Hospital's rule.
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\dfrac{\frac{2}{n}}{1}$
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\dfrac{2}{n}$
$\lim\limits_{n \to \infty}a_n=\dfrac{2}{\infty}$
$\lim\limits_{n \to \infty}a_n=0$
Therefore, the sequence converges to $0$.
