Answer
converges to $0$
Work Step by Step
Let $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} [\dfrac{(\ln n)^{5}}{\sqrt n}]$
Here, $ \lim\limits_{n \to \infty} [\dfrac{(\ln n)^{5}}{\sqrt n}]=\dfrac{\infty}{\infty}$
This shows that the limit has an Indeterminate form so, we will use L-Hospital's rule.
This implies that $\lim\limits_{n \to \infty} [ \dfrac{5(\dfrac{\ln n)^{4}}{n})}{\dfrac{1}{2}( \sqrt n)}]=\lim\limits_{n \to \infty} \dfrac{(5)(2) (\ln n)^4}{\sqrt n}=\dfrac{\infty}{\infty}$
Now, use L-Hospital's rule again.
Then, $\lim\limits_{n \to \infty} \dfrac{(5!) (2^5)}{\sqrt n}=0$
Hence, {$a_n$} is Convergent and converges to $0$.
