The series given is:

$\sum_{n=1}^\infty \frac{n^n}{n!}.$

To determine whether the series converges or diverges, let us analyze the terms using standard convergence tests.

Step 1: Examine the general term

The general term of the series is: $a_n = \frac{n^n}{n!}.$

Step 2: Apply the ratio test

The ratio test involves finding: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.$ Substitute $a_n = \frac{n^n}{n!}$: $\frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)^{n+1}}{(n+1)!}}{\frac{n^n}{n!}} = \frac{(n+1)^{n+1} \cdot n!}{n^n \cdot (n+1)!}.$ Simplify: $\frac{a_{n+1}}{a_n} = \frac{(n+1)^{n+1}}{n^n \cdot (n+1)} = \frac{(n+1)^n \cdot (n+1)}{n^n \cdot (n+1)} = \frac{(n+1)^n}{n^n}.$ Simplify further: [ \frac{a_{n+1}}{a_n} = \left(\frac{n+1}{