Rank the listed functions by order of growth.

$n^2$

$\frac{n}{logn}$

$nlogn$

$(1\mathrm{.}001{)}^n$

$\frac{1}{n^2}$

$lo{g}^{100}n$

$n!$

$n^{lglgn}$

That is$,$ find an arrangement $f_1,f_2,$dots,$f_8$ of the functions satisfying

$f_1=O(f_2)$

$f_2=O(f_3)$

vdots

$f_7=O(f_8)$

Justify your ordering.

Note that $lo{g}^{k}n$ is the usual way of writing $(logn{)}^k.$