Answer
Converges to $0$
Work Step by Step
Consider $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} [\dfrac{(\ln n)^{5}}{\sqrt n}]$
Since, $ \lim\limits_{n \to \infty} [\dfrac{(\ln n)^{5}}{\sqrt n}]=\dfrac{\infty}{\infty}$
Need to apply L-Hospital's rule.
So, $\lim\limits_{n \to \infty} [ \dfrac{5(\ln n)^{4}/n}{1/2 \sqrt n}]=\lim\limits_{n \to \infty} \dfrac{(5)(2) (\ln n)^4}{\sqrt n}]=\dfrac{\infty}{\infty}$
Again apply L-Hospital's rule.
we have $\lim\limits_{n \to \infty} \dfrac{5! \cdot 2^5}{\sqrt n}=0$
Hence, $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is Convergent and converges to $0$
