 Uniform depletion Markov chain

A finite stock decreases by choosing a uniformly random positive number of its remaining items at every step. The remaining stock is a Markov chain with $P(i,j)=1/i$ for $0\le j<i$ and $P(0,0)=1$. State $0$ is an absorbing state, and every positive state is a transient state. If $e_i$ denotes the expected number of positive-stock steps before absorption, a first-step decomposition gives $e_0=0$ and $i{e}_i=i+{\sum }_{j<i}{e}_j$. Subtracting consecutive equations yields $e_i-e_{i-1}=1/i$, so $e_i=H_i$, the $i$th harmonic number.