 Solution

Let $e_k$ be the expected number of future children receiving stickers from a box containing $k$, with $e_0=0$. Conditioning on the first uniform remainder givesSubtract the corresponding identity for $k-1$ after multiplying each by its denominator. This gives $k{e}_k=k{e}_{k-1}+1$, so $e_k=e_{k-1}+1/k$. Therefore $e_N={\sum }_{j=1}^{N}1/j$, andThe recurrence counts the current positive recipient once and excludes all later children arriving after the box is empty.