Refer to Section 6.2 and show that, if the occupation number \(n_{\varepsilon}\) of an energy level \(\varepsilon\) is restricted to the values \(0,1, \ldots, l\), then the mean occupation number of that level is given by

\[
\left\langle n_{\varepsilon}ightangle=\frac{1}{z^{-1} e^{\beta \varepsilon}-1}-\frac{l+1}{\left(z^{-1} e^{\beta \varepsilon}ight)^{l+1}-1} .
\]

Check that while \(l=1\) leads to \(\left\langle n_{\varepsilon}ightangle_{\text {F.D. }}, l ightarrow \infty\) leads to \(\left\langle n_{\varepsilon}ightangle_{\text {B.E. }}\).