Chapter 9, Problem 24RE

It can be proved that

$\underset{n\to +\infty }{\lim }\sqrt[n]{n!}=+\infty \text{and}\underset{n\to +\infty }{\lim }\frac{\sqrt[n]{n!}}{n}=\frac{1}{e}$

In each part, use these limits and the root test to determine whether the series converges.

$\begin{array}{l}\text{(a)}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{2^k}{k!}}\\ \text{(b)}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{=k^k}{k!}}\end{array}$