= Dyadic spherical shells with inverse-radius density scaling
{title2=$M(2^{-N})=M(1)\,4^{-N}$}
Suppose the shell $2^{-n}<r<2^{1-n}$ has <mass density> $2^{n-1}\rho_0$. Its mass is $(14\pi\rho_0/3)4^{-n}$, by integrating $4\pi r^2\rho\,dr$. Summing the <geometric series> gives total mass $14\pi\rho_0/9$. At $r=2^{-N}$ the enclosed shells have indices $n\geq N+1$, so the enclosed mass is the total mass times $r^2$. The magnitude of the <Newtonian gravitational field> at every such boundary is therefore the same, $14\pi G\rho_0/9$. This does not assert that the field is constant throughout each shell.
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