each succeeding term xn+1 is produced from the preceding term xn by multiplying by the factor 1/r, then the sum of all the terms xn+1 + xn+2... is equal to xn * 1/(r-1).
So, I ran a Python simulation and it turns out to be correct!
If I'm reading the article correctly, this turns out to be a simple result from the theory of geometric series, and it looks like it works even for fractional r. Modifying the code above to test fractional r shows this is true. Department of "learn something new every day"....
My guess is that von Neumann saw all three pieces instantly:
• the first term and step size of the series
• the formula for the terms after the first one
• the easy way