term of the expansion.\
The same procedure just described to expand the integral $\mathcal{A}_d(z_{\textrm{max}},0)$ can be applied to the integral (\[L(zmax)\]) as well, obtaining $L$ as an expansion in powers of $z_0- z_{\textrm{max}}$. It is then useful to compare the divergences of these two quantities as $z_{\textrm{max}} \rightarrow z_0$ in order to see how the finite term of the entanglement entropy scales with the width of the strip, and therefore with the volume.

Charged black hole {#section nh charged bh}
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In this section we apply the method just described to the charged black hole in $AdS_{d+2}$ in its three different regimes of neutrality, extremality and non extremality. The metric and its properties are reviewed in the appendix \[app chargedBH\].\
The metric of the charged black hole in $AdS_{d+2}$ reads $$\label{charged BH metric z}
\frac{ds^2}{R^2}\,=\,\frac{-f dt^2+ d\vec{x}^2}{z^2}+\frac{dz^2}{f z^2}
\hspace{1.6cm}
f\,=\,1+Q^2\left(\frac{z}{R^2}\right)^{2d}-M\left(\frac{z}{R^2}\right)^{d+1}$$ where $M$ is the mass and $Q$ is the charge of the black hole. The radial direction is parameterized by $z$ and the boundary is at $z=0$. The position $z_0$ of the horizon is given by the smallest zero of the emblacking function $f(z)$. Since the metric (\[charged BH metric z\]) falls into the class of metrics described by (\[ds20 