Answer
Divergent
Work Step by Step
$\sum_{n=0}^{\infty}\arctan^n(n)$
By Root Test, we have
$\lim_{n\to\infty}\sqrt[n]{\biggl|\arctan^n(n)\biggr|}=\lim_{n\to\infty}\biggl|\arctan^n(n)\biggr|=\frac{\pi}{2}$
Since the limit $ > 1$, the series is divergent by Root Test