The floor that correlated failure puts under redundancy

Model failures with a common-cause fraction $\beta$: with probability $\beta$ an event takes down all $n$ replicas together, otherwise they fail independently, each with unavailability $U$.

1. Derive $U_n = (1-\beta)U^n + \beta U$ and show it is decreasing in $n$ with limit $\beta U$.
2. With $U = 10^{-3}$, tabulate $U_n$ for $n = 1,2,3,\infty$ at $\beta = 0$ and $\beta = 0.01$.
3. Find the smallest $n$ at which adding a further replica improves $U_n$ by less than 10% relative, when $\beta = 0.01$ and $U = 10^{-3}$.
4. State what this implies about where availability effort should go.