# Black Holes and the Expanding Universe

Calculus Level 5

In a universe whose geometry can be represented by a spherical shell, two black holes form every quantum of time and the universe expands such that its radius is $$\Bigg(\displaystyle \sum_{n=0}^t \frac {(-1)^n}{2n+1} {t \choose n}\Bigg)^{-1}$$ units at t quanta of time from the beginning (i.e. when t = 0). If the universe originally had 1 black hole and the eventual density of black holes in the universe is $$\rho$$ black holes per square unit, find the value of $$\frac {1}{\rho}$$.

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