Uniform decreasing Markov chain
= Uniform decreasing Markov chain
{title2=$P_{ij}=1/(i-1)\ (1\le j<i)$}
On $\{1,\ldots,n\}$, make $1$ an <absorbing state> and let each state $i>1$ move uniformly to $\{1,\ldots,i-1\}$. Its <expected hitting time> of $1$, allowing zero time when started at $1$, is $e_i=H_{i-1}$. <First-step analysis> gives $e_i=1+(i-1)^{-1}\sum_{j<i}e_j$; subtracting consecutive recurrences yields $e_i-e_{i-1}=1/(i-1)$ and hence the formula.