Multiplicatively Decreasing I

Question

You play the following game starting on turn $i = 1$: Initialize turn to $i = 1$. Generate $X_i \sim \unif{0}{\alpha^{i-1}}$, where $0 < \alpha < 1$, independently of other random variables. You can choose to either receive a payout of $X_i$ (the game ends), or you can choose to reject $X_i$. If you reject $X_i$, $i$ gets incremented to $i+1$, and the process goes back up to the second step. Assuming you play optimally, what is your expected payout from this game? Report the answer when $\alpha = 1/2$ to the nearest thousandth.