Answer
Diverges
Work Step by Step
Consider $u_n=(-1)^{n+1}\dfrac{n^2+5}{ n^2+4}$
Now $\lim\limits_{n \to \infty} u_n=\dfrac{n^2+5}{ n^2+4}=1$
$\lim\limits_{n \to \infty} (-1)^{n+1}\dfrac{n^2+5}{ n^2+4}=DNE$
Thus the series diverges by the nth Term Test for Divergence.
