Answer
Converges to $1$
Work Step by Step
Consider $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{\ln n}{\ln 2n}$
But $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln 2n}=\dfrac{\infty}{\infty}$
Need to apply L-Hospital's rule.
So, $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln 2n}=\lim\limits_{n \to \infty} \dfrac{1/n}{2/2n}$
or, $=1$
Hence, $\lim\limits_{n \to \infty} a_n=1 $ and {$a_n$} is convergent and converges to $1$.
