Answer
Diverges
Work Step by Step
Let us consider $a_n=(-1)^{n+1}\dfrac{n^2+5}{ n^2+4}$ and $u_n$ refers to all positive values of $n$.
Here, $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty}\dfrac{n^2+5}{ n^2+4}=\lim\limits_{n \to \infty}\dfrac{1+5/n^2}{1+4/n^2}=1$
Also, $\lim\limits_{n \to \infty} (-1)^{n+1}\dfrac{(n^2+5)}{ (n^2+4)}=DNE$
Hence, the series diverges by the Nth Term Test for Divergence.
