2022 Putnam A4

Suppose that $X_1, X_2, \dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \infty$ if there is no such integer. Find the expected value of $S$.

假设 $X_1, X_2, \dots$ 是独立且均匀随机选择的 0 到 1 之间的实数。设 $S = \sum_{i=1}^k X_i/2^i$，其中 $k$ 是满足 $X_k < X_{k+1}$ 的最小正整数，如果不存在这样的整数，则 $k = \infty$。求 $S$ 的期望值。