Answer
Convergent
Work Step by Step
Alternating series test:
Suppose that we have series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$.
Then if the following two condition are satisfied the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
In the given problem, $b_{n}=\frac{n^{2}-1}{n^{3}+1}$
1. $b_{n}=\frac{n^{2}-1}{n^{3}+1}$, is a dreasing sequence.
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{n^{2}-1}{n^{3}+1}$
$=0$
Hence, the series is convergent by AST.
