Answer
Divergent
Work Step by Step
$\lim\limits_{n \to \infty}a_{n}=\lim\limits_{n \to \infty}(-1)^{n}\frac{n^{2}}{n^{2}+n+1}$
Divide both numerator and denominator by $n^{2}$
$=\lim\limits_{n \to \infty}(-1)^{n}\frac{\frac{n^{2}}{n^{2}}}{\frac{n^{2}+n+1}{n^{2}}}$
$=\lim\limits_{n \to \infty}(-1)^{n}\frac{1}{1+\frac{1}{n^{2}}+\frac{1}{n^{2}}}$
$=\lim\limits_{n \to \infty}(-1)^{n}\frac{1}{1+0+0}$
$=\lim\limits_{n \to \infty}(-1)^{n}\ne 0$
which means that the series diverges by the Test of Divergence.
