Answer
The series converges.
Work Step by Step
We have the given series
$\sum_{n=1}^{\infty} \frac{n^{2}-n}{n^{5}+n}=\sum_{n=1}^{\infty} \frac{n-1}{n^{4}+1}$
We apply the limit comparison test with $b_n=(\frac{1}{n^3})$ (a convergent p-series with $p=3\gt 1$):
$L=\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=\lim_{n\rightarrow\infty}\frac{n-1}{n^{4}+1}\times(n^3)=\lim_{n\rightarrow\infty}\frac{n^4-n^3}{n^{4}+1}=\lim_{n\rightarrow\infty}\frac{1-1/n}{1+1/n^4}=1$
Since $L=1$, our starting series converges.
