Which is greater: $1000^{1000}$ or $1001^{999}$
Look at the quotient $$ \frac{1001^{999}}{1000^{1000}}=\frac1{1001}\underbrace{\left(1+\frac1{1000}\right)^{1000}}_{\approx e}$$
$\forall r \in \Bbb N-\{1\}$, we have by applying the AM-GM inequality to the $r$ numbers $r-1$ of which equal $r+1$ and one $1$, we have, $$\frac {1+(r-1)(r+1)}{r} \gt (1 \times (r+1)^{r-1})^\frac {1}{r}$$ wherefrom we have, $r^r \gt (r+1)^{r-1}.$