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.
Given: $\Sigma_{n=1}^{\infty}(-1)^{n}\frac{ n}{\sqrt {n^{3}+2}}$
In the given problem, $b_{n}=\frac{ n}{\sqrt {n^{3}+2}}$
which satisfies both conditions of Alternating Series Test as follows:
1. $b_{n}=\frac{ n}{\sqrt {n^{3}+2}}$
$=\frac{ n}{\sqrt {n^{2}(n+\frac{2}{n^{2}})}}$
$=\frac{ n}{n\sqrt {n+\frac{2}{n^{2}}}}$
$=\frac{ 1}{\sqrt {n+\frac{2}{n^{2}}}}$
therefore, $b_n$ is decreasing because the denominator is increasing.
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{ 1}{\sqrt {n+\frac{2}{n^{2}}}}$
$=0$
Hence, the given series is convergent by Alternating Series Test.
