= Solution
The <ratio test> says that a <series> $\sum u_n$ whose terms are eventually nonzero converges absolutely if $\limsup |u_{n+1}/u_n|<1$; in particular, it suffices to have $|u_{n+1}/u_n|\leq q<1$ eventually, since then its terms are bounded by a convergent <geometric series>. The <comparison test for series> states that, for nonnegative terms, comparison with a larger convergent series gives convergence, and comparison with a smaller divergent series gives divergence. For $u_n=n!/n^n$,
$$
\frac{u_{n+1}}{u_n}=\left(\frac n{n+1}\right)^n.
$$
By the <binomial theorem>, $(1+1/n)^n\geq1+n(1/n)=2$. Thus $u_{n+1}/u_n\leq1/2$ for every $n\geq1$, and induction gives $0<u_n\leq2^{1-n}$. The <comparison test for series> with $\sum2^{1-n}=2$ proves
$$
\boxed{\sum_{n\geq1}\frac{n!}{n^n}\text{ converges absolutely}.}
$$
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