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