Imagine an infinite series of nested hollow shells of equal thickness around Earth.
The number of stars in each shell is quadratic relative to the radius. The brightness of each star is inversely quadratic relative to the radius of the shell. Thus, each shell has a constant total brightness, as perceived from Earth. An infinite number of these shells add up to infinite brightness.
I'm not quite following. The light in each "shell" continues to dissipate in an inversely quadratic method from the shells distance from earth, no? It would help if I had this drawn out.
No, the total light in each spherical "shell" is the same as every other shell.
Stars that are 20 light-years away are 4x dimmer than stars 10 light-years away. But there are 4x more stars 20 light-years away than 10 light-years away. As you go outward from Earth, the stars get dimmer and more numerous, but these balance each other, and the infinite sum is 1 + 1 + 1 + ...
(This is no chance happening; regardless of the # of spatial dimensions the universe has, this relationship holds true.)
The number of stars in each shell is quadratic relative to the radius. The brightness of each star is inversely quadratic relative to the radius of the shell. Thus, each shell has a constant total brightness, as perceived from Earth. An infinite number of these shells add up to infinite brightness.