I was trying to come up with a way to show that $\sum_{i=0}^{n}i^i < cn^n$, where $c$ is some positive constant.
I figured if this were true:
$\sum_{i=1}^{n-1}i^i < n^n, n>1$
in other words:
$1^1 + 2^2 + ... + (n-1)^{(n-1)} < n^n$
then the first statement must also be true (for example, when $c\ge2$).
It seems like the latter statement is true, but how can one prove it? Also, do these numbers of the form $x^x$ have a special name?