= Solution
Use the luminosity convention in the supplied flux relation: $P_\nu=L_\nu/(4\pi)$, where $L_\nu$ is the total <spectral luminosity>. For a general spectrum $P_\nu\propto\nu^\alpha$, the <cosmological spectral flux-density relation> gives
$$
S_{\nu_0}=\frac{(1+z)P_{(1+z)\nu_0}}{d_L^2}
=\frac{P_{\nu_0}(1+z)^{\alpha-1}}{r_e^2}.
$$
In a spatially flat universe, a shell subtending the <solid angle> $d\Omega$ contains $dN=n_0r_e^2dr_e\,d\Omega$ sources, because $n_0$ is their <comoving number density>. The <cosmological spectral background> intensity per unit <solid angle> is consequently
$$
I_{\nu_0}(z_{\max})
=n_0P_{\nu_0}\int_0^{r_e(z_{\max})}(1+z)^{\alpha-1}dr_e
=\frac{cn_0P_{\nu_0}}{H_0}
\int_0^{z_{\max}}(1+z)^{\alpha-5/2}\,dz.
$$
This is the <cosmological background intensity from comoving emissivity> specialized to identical, nonevolving sources in an <Einstein-de Sitter universe>. For $\alpha=1$ the integrand reduces to $(1+z)^{-3/2}$, or directly $dI_{\nu_0}=n_0P_{\nu_0}dr_e$. The finite <comoving particle horizon> therefore gives
$$
I_{\nu_0}(z_{\max})=
\frac{2cn_0P_{\nu_0}}{H_0}[1-(1+z_{\max})^{-1/2}],
\qquad
\boxed{I_{\nu_0}(\infty)=\frac{2cn_0P_{\nu_0}}{H_0}.}
$$
For $\alpha=2$, instead,
$$
\boxed{I_{\nu_0}(z_{\max})=
\frac{2cn_0P_{\nu_0}}{H_0}[\sqrt{1+z_{\max}}-1]\longrightarrow\infty.}
$$
The emitted <frequency> grows with <redshift>, and the rising <spectral luminosity> now defeats the redshift dimming. The <Einstein-de Sitter spectral-background convergence criterion> is $\alpha<3/2$; equality produces a logarithmic divergence. This divergence describes the unbounded power-law, eternal-population model. A finite formation epoch, a high-frequency spectral break or absorption changes that model and can make its background finite.
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