Rate of convergence of $r_n = n^x$ dependent on the parameter $x$.
$r_{n+1}=(n+1)^x = n^x+xn^{x-1}+o(n)$
$lim_{n\to \infty}\frac{r_{n+1}}{r_n} = lim_{n\to \infty}\frac{n^x+xn^{x-1}+o(n)}{n^x} = lim_{n\to \infty}1+\frac{x}{n}+o(n)/{n^x} = 1$
So the rate of convergence is sublinear in both $x \in (0,1)$; $x \ge 1$ and $x < 0$ cases?