Use the Root Test to determine whether the series $\sum\limits_{k\,=\,1}^{ \infty }\left( {\dfrac{8k+3}{5k-2}}\right) ^{\!\!k}$ converges or diverges.

Solution $\sum\limits_{k\,=\,1}^{\infty }\left( {\dfrac{8k+3}{ 5k-2}}\right) ^{\!\!k}$ is a series of nonzero terms. The $n$th term is $a_{n}=$ $\left( {\dfrac{8n+3}{5n-2}}\right) ^{\!\!n}\!$. Since $a_{n}$ involves an $n$th power, we use the Root Test. $$\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\,\vert a_{n}\vert }=\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\left( {\dfrac{8n+3}{5n-2}} \right) ^{\!\!n}}=\lim\limits_{n\,\rightarrow \,\infty }\frac{8n+3}{5n-2}=\frac{8}{5}>1$$

The series diverges.