Answer
converges to $0$
Work Step by Step
Let $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} [\dfrac{(\ln n)^{200}}{n}]$
Here, $ \lim\limits_{n \to \infty} [\dfrac{(\ln n)^{(200)}}{n}]=\dfrac{\infty}{\infty}$
This shows that the limit has an Indeterminate form so, we will use L-Hospital's rule.
This implies that $\lim\limits_{n \to \infty} \dfrac{(200) (\ln n)^{(199)}}{n}=\dfrac{\infty}{\infty}$
Now, use L-Hospital's rule again
Thus, $\lim\limits_{n \to \infty} \dfrac{(200) !}{n}=0$
Hence, {$a_n$} is Convergent and converges to $0$
