I've recently stumbled upon an alternative version of showing Olbers' Paradox analytically, namely the following problem from Arfken and Weber's Mathematical Methods for Physicists:
Assume a static universe in which the stars are uniformly distributed. Divide all space into shells of constant thickness; the stars in any one shell by themselves subtend a solid angle of $ω_0$. Allowing for the blocking out of distant stars by nearer stars, show that the total net solid angle subtended by all stars, shells extending to infinity, is exactly $4π$.
Now, I am familiar with the light intensity formulation of the paradox, but here I seem to have a bit of a problem working with solid angles. This is my attempt at the solution:
I assumed that every star on any spherical shell covers some constant fraction of the surface, call it $dS$. Then the solid angle that each star in some shell subtends is $d\omega_0(r)=\frac{dS}{r^2}$. Hence, for every spherical shell, the total sum of solid angles should be:
$$N_i \ d\omega_0=4\pi r_i^2dr_i \gamma \frac{dS}{r_i^2}=4\pi\gamma dr_i dS$$
where $N_i$ is the number of stars/shell, $4\pi r_i^2dr_i$ is the volume of the shell with thickness $dr_i$ and $\gamma$ is the number density of stars/volume of the shell. Now summing over all shells (and all stars, in essence), we have
$$\omega_{total}=Nd\omega_0=4\pi\gamma dS \int_0^\infty dr$$
where $N$ is the number of all stars and also noting $\gamma=\frac{N}{\iiint_V drdS}$, since the distribution is uniform. But this logic gets me nowhere and I think the reason is that I do not fully understand the question, namely what exactly is meant by
the stars in any one shell by themselves subtend a solid angle of $ω_0$. Allowing for the blocking out of distant stars by nearer stars
Also, this problem is under the chapter of convergence of series in the previously mentioned book, thus I assume these should pop up somewhere, but I can't see any apart from the sum over all shells.
Any help is much appreciated!