Answer
Convergent
Work Step by Step
Use Limit Comparison Test with $a_{n}=\frac{5+2n}{{(1+n^{2})^{2}}}$ and $b_{n}=\frac{1}{n^{3}}$
$\lim\limits_{n \to \infty}\frac{a_{n}}{b_{n}}=\lim\limits_{n \to \infty}\frac{n^{3}(5+2n)}{(1+n^{2})^{2}}$
$=\lim\limits_{n \to \infty}\frac{5n^{3}+2n^{4}}{(1+n^{2})^{2}}$
$=\lim\limits_{n \to \infty}\frac{\frac{5}{n}+2}{(\frac{1}{n^{2}}+1)^{2}}$
$=2\gt 0$
$\Sigma_{n=1}^{\infty}\frac{1}{n^{3}}$ is convergent because a $p-$series with $p=3 \gt 1$ is convergent, thus the series $\Sigma_{n=1}^{\infty}\frac{5+2n}{{(1+n^{2})^{2}}}$ also converges.
