In *Introduction to Solid State Physics*, eighth edition, by Kittel, page 141, eqs. (20,21), the density of states for electron in conduction in three dimensions is 
$$D(\epsilon)\equiv \frac{dN}{d\epsilon}=\frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)\epsilon^{1/2}=\frac{3}{2} \frac{N}{\epsilon}.$$

The same argument could apply such that in two dimensions
$$D(\epsilon)= \frac{2}{2}\frac{N}{\epsilon},$$
and in one dimension
$$D(\epsilon)= \frac{1}{2}\frac{N}{\epsilon}$$
—i.e. with each increase of dimension, the density of states increase by $\frac{1}{2}\frac{N}{\epsilon}$.

1. Why does each dimension increase the density of states $D(\epsilon)$ by $\frac{1}{2}\frac{N}{\epsilon}$?

2. What's the order of $D(\epsilon)$ in 3D, $\epsilon^{1/2}$ or $\epsilon^{-1}$?