Chapter 5 - Series Solutions of Second Order Linear Equations - 5.1 Review of Power Series - Problems - Page 249: 8

$\rho =e$

$\sum_{\infty }^{n=1}(\frac{n!x^n}{n^n})$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\rightarrow \;\;\;\;\;\;\; \lim_{n\rightarrow \infty} \left | \frac{(n+1)!x^{(n+1)}}{(n+1)^{(n+1)}}\;.\;\frac{n^n}{n!x^n} \right | = \lim_{n\rightarrow \infty} \left | x\frac{n^n}{(n+1)^n} \right | = |\frac{x}{e}|$ $|\frac{x}{e}|<1\;\;\;\;\;\;\;\;\;$ $\;\;\;\;\;\;\;\;\;|x|< e$ $\rho =e$