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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