 Uniform decreasing Markov chain

On $\{1,\dots ,n\}$, make $1$ an absorbing state and let each state $i>1$ move uniformly to $\{1,\dots ,i-1\}$. 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.