Using the Root Test

Use the Root Test to determine whether the series \(\sum\limits_{k\,=\,1}^{ \infty }\dfrac{e^{k}}{k^{k}}\) converges or diverges.

Solution \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{e^{k}}{k^{k}}\) is a series of nonzero terms. The \(n\)th term is \( a_{n}=\) \(\dfrac{e^{n}}{n^{n}}=\left(\dfrac{e}{n}\right)^{n}\). Since \(a_{n}\) involves an \(n\)th power, we use the Root Test. \[ \lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\,\vert a_{n}\vert }=\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\left( \frac{e}{n}\right) ^{n}}=\lim\limits_{n\,\rightarrow \,\infty }\frac{e}{n}=0<1 \]

The series \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{e^{k}}{k^{k}}\) converges.