Density of Energy Levels - Strange Summation

[4piElliot0]

Homework Statement

*This is not my whole problem, I am only stuck on how to interpret one part of the question. Put simply, I want to find the expression for the density of energy levels in a given energy band per unit volume (in some crystal structure). Say I have an infinitesimal interval of energy levels $\Delta$, I would like to find the number of energy levels per $\Delta$ and unit volume.

Homework Equations

I am given the expression $$ g_n(E) = \frac{1}{ V \Delta } \sum_{\mathbf{k}, E < E_{n\mathbf{k}} < E + \Delta} 1 $$ which I write as
$$ g_n(E) = \frac{1}{ V \Delta } \sum_{\mathbf{k}} \sum_{E < E_{n\mathbf{k}} < E + \Delta} $$

The Attempt at a Solution

Using the periodic boundary conditions I can convert the sum over $\mathbf{k}$ into an integral in reciprocal space,
$$\sum_{\mathbf{k}} \rightarrow \frac{V}{(2 \pi)^3} \int d^3k $$
at which point I am stuck on how to interpret the second sum with index $E < E_{n\mathbf{k}} < E + \Delta$. I was thinking about taking the limit that $\Delta \rightarrow 0$ the sum would become an integral,
$$ \frac{1}{\Delta} \sum_{E < E_{n\mathbf{k}} < E + \Delta} \rightarrow \int dE $$
but this doesn't seem quite right. I am stuck on how to deal with that sum, and what the bounds of the integral should be.
Any help with that step would be much appreciated!

 

[mfb]

E and k are related, I don't think you can separate them like that. The number of energy levels within that infinitesimal energy interval will depend on k, and it is some external input you'll need.