Optimal Stopping for Uniform Draws

You draw values one at a time from $U[0,1]$, up to $n$ draws total. After each draw, you see the value and must decide: keep it (and stop), or throw it away and continue to the next draw. If you reach the $n$-th draw, you must keep it regardless. What stopping strategy maximizes your expected payoff? Derive the expected value under the optimal strategy as a function of $n$, and compute it for small $n$.