An interesting cosmology website proposes to calculate the magnitude of the anti-gravity force driving the accelerating expansion of the universe (a.k.a. “dark energy”) by a simple manipulation of Newton’s gravity formula. They presume to calculate the magnitude of dark energy by (and I quote):

“calculating how much energy is required to expand a 2-body system at the observed Hubble rate. Then we repeat the calculation for an expanding 3-body system; then a 4-body; then an 8-body. From that, we extrapolate to the infinite-body system. Once we have a formula for calculating the expansion of an infinite system, we can plug in the numbers from our universe…and get an idea of the magnitude of the mysterious energy driving the expansion.”

They assume a homogenous, isometric universe where clumps of matter—galaxy superclusters—are of a uniform mass $\displaystyle M$ and distance $\displaystyle R$ apart from each other. (It is immaterial to the present challenge how they derive these values). The universe is thus modeled as a cube of side $\displaystyle R$ repeating itself in all directions infinitely, with clumps of matter (galaxy super-clusters) at each of the interstices of the cubes, each of uniform mass $\displaystyle M$. They plug this uniform distance and mass into Newton’s gravity formula.

$\displaystyle F=\frac{GM^2}{R^2}$

Then they plug this into the formula Power = velocity x force, i.e.:

$\displaystyle P=vF=\frac{vGM^2}{R^2}$

For velocity they substitute “HR”—the Hubble constant times the uniform distance between clumps of matter, resulting in:

$\displaystyle P=\frac{HGM^2}{R}$

Finally, they stress the need to calculate "relative power" as opposed to absolute power. The formula for relative power, they say, is just power divided by the total mass of the system:

$\displaystyle P_{rel}=\frac{g(n)P}{f(n)M}=\frac{g(n)HGM}{f(n)R}$

(where $\displaystyle g(n)$ is the formula that gives the sum of lengths of unit R in the $\displaystyle n^3$ system, and $\displaystyle f(n)$ is the formula that gives the total mass of the system.)

However, they do not actually arrive at the desired “formula for calculating the expansion of an infinite system.” They take their calculation only up to a 1 cube (i.e. n=1), 8-body system, for which they get a value of

$\displaystyle \frac{(12+12\sqrt{2}+4\sqrt{3})HGM}{8R}=4.5\frac{H GM}{R}$

They then say:

“So the answer is between 4.5-times-HGM/R and infinity. That narrows it down!... [R]eaders…will recognize this as an ‘infinite series’ problem…. There are two parts to this type of problem: 1) Does the series 'converge' to a finite number? 2) If so, what does it converge to?... Since the staff at [this website] is not mathematically gifted, we will make an educated guess at the answer, and invite our more mathematically talented readers to provide the correct answer, if the guess is wrong. In that event, corrections will be made."

Their guess is that it converges, and that it converges to $\displaystyle 8\pi$. Their only basis for this guess is that $\displaystyle 8\pi$ is the factor that appears in Einstein’s general relativity equation.

Let’s help the staff at this website make their theory a little more rigorous. The challenge here is to prove or disprove their guess that

$\displaystyle \lim_{n\to \infty }\frac{g(n)HGM}{f(n)R}=8\pi\frac{HGM}{R}$

AND to point out any mathematical errors in the chain of logic leading up to their guess.

Note:- There are probably a few questionable leaps of logic that are not strictly mathematical. Let’s leave those aside and focus exclusively on any mathematical fallacies.

My answer has been forwarded to all moderators.