Answer
Diverges
Work Step by Step
Let us consider $p_n=(-1)^{n+1}\dfrac{3+n}{5+n}$ and $p_n$ refers to all positive values of $n$.
We can see that $\lim\limits_{n \to \infty} p_n=\lim\limits_{n \to \infty}\dfrac{1+\dfrac{3}{n}}{1+\dfrac{5}{n}}=1\ne 0$
Thus, the given series diverges by Nth Term Test