Answer
converges to $1$.
Work Step by Step
Let $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{\ln n}{\ln 2n}$
But $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln 2n}=\dfrac{\infty}{\infty}$
This shows that the limit has an Indeterminate form so we will use L-Hospital's rule.
This implies that $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln 2n}=\lim\limits_{n \to \infty} \dfrac{\dfrac{1}{n}}{\dfrac{2}{2n}}=1$
Thus, $\lim\limits_{n \to \infty} a_n=1 $ and {$a_n$} converges to $1$.