Use the root test to determine whether the series $\displaystyle{\sum_{n=1}^{\infty} {\left(\frac{6n^{3}-n-2}{4n^{2}+n+8}\right)^{n}}}$ converges or diverges. Since $\displaystyle{\lim_{n \to \infty} \bigg(}$ $\displaystyle{\bigg) = }$ , which is choose less than 1 equal to 1 greater than 1 , the series choose converges diverges may converge or diverge by the root test.