## Calculus 8th Edition

$\lim\limits_{n \to \infty}a_{n}=\lim\limits_{n \to \infty}(-1)^{n}\frac{n^{2}}{n^{2}+n+1}$ 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.