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.