I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N R(N,k)$.

I'm guessing this is a well known distribution, but I don't know how to search for it.

Update: The motivation is the following story. There are $N$ people, and $N$ items. Each person has a randomly uniformly drawn preference order over the $N$ items. The first person gets his favorite item, the second gets his favorite item out of what's left after the first person took his item, and so on, the $k$-th person gets his favorite item out of what's left after $1,..,k-1$ took theirs. So $R(N,N-k)-1$ is the expected number of items that person $k$ wanted but were already gone.

Thanks!