Answer
Diverges
Work Step by Step
Let us suppose that a series $\Sigma a_n $ and $a_n=(-1)^n b_n$ or, $a_n=(-1)^{n+1} b_n$ with $b_n \geq 0$ for all $n$. We will apply the Alternating Series Test to compute when an infinite series converges, it must follow the following two conditions:
a) $\lim\limits_{n \to \infty}b_n=0$
b) $b_n$ shows a decreasing sequence.
Here, we have the $b_n=\dfrac{n}{n-3}$
$\lim\limits_{n \to \infty}b_n=\lim\limits_{n \to \infty} \dfrac{n}{n-3}=\lim\limits_{n \to \infty} \dfrac{1}{1-3/n}=1 \ne 0$
This means that the first condition for the Alternating Series Test fails. Thus, the given series diverges.
