Using the Root Test

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.