Answer
The series is absolutely convergent.
Work Step by Step
Use comparison test with $a_n=\frac{arctan n }{n^{2}}$ and $b_{n}=\frac{1}{n^{2}}$
$\lim\limits_{n \to \infty}\frac{a_{n}}{b_{n}}=\lim\limits_{n \to \infty}\frac{arctan n/n^{2} }{n^{2}}$
$=\lim\limits_{n \to \infty}arctan n$
$=\frac{\pi}{2}\ne 0 \ne \infty$
The given series absolutely converges because $\Sigma_{n=1 }^{\infty}\frac{1}{n^{2}}$ is a convergent p-series with $p=2\gt 1$
Hence, the series is absolutely convergent.