Answer
converges to $\dfrac{\pi}{2}$
Work Step by Step
As we know that $ \lim\limits_{n \to \dfrac{\pi}{2}} \tan x=\dfrac{\pi}{2}$
Let $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \tan^{-1} n$
This implies that $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \tan^{-1} n$
and $a_n=\dfrac{\pi}{2}$
Thus, {$a_n$} is Convergent and converges to $\dfrac{\pi}{2}$.