I am reading Höhle & Held's paper, "Bayesian Estimation of the Size of a Population" (https://epub.ub.uni-muenchen.de/2094/1/paper_499.pdf) which discusses the well-known problem consisting in estimating the unknown size $N$ of a population whose individuals are numbered from $1$ to $N$, given the maximum number $x$ observed on a sample of size $n$. It’s known that, knowing $N$, the probability of observing $x$ as the maximum is (their formula $1$): $$P(x|N)=\frac{ \begin{pmatrix}x-1\\n-1\end{pmatrix} }{ \begin{pmatrix}N\\n\end{pmatrix} }, \quad n \leq x \leq N$$ Assuming a prior distribution $P(N)$ for $N$, the authors then write as follows the posterior distribution of $N$ (their formula $3$): $$P(N|x)= \frac{P(x|N)P(N)}{ \sum_{N’=x}^\infty {P(x|N’)P(N’)} }$$ That is to say, they simply use the Bayes’ theorem. So far so good.
Where it gets stuck is when they choose for prior of $N$ an improper uniform prior on all positive integers, which they write $P(N) \propto 1$, for $N=0,...,\infty$, which is obviously not a probability distribution. They then deduce (verbatim): “under the assumption of an improper uniform prior, the posterior distribution ($3)$ simplifies to (now their formula $7$):” $$P(N|x)=\frac{ {\begin{pmatrix}N\\n\end{pmatrix}}^{-1} }{ \sum_{N’=x}^\infty {\begin{pmatrix}N'\\n\end{pmatrix}}^{-1} }$$ But I don't understand at all how it “simplifies to”... In order to go from formula $3$ to formula $7$, it looks like they took $P(N)=P(N’)=\frac{1}{\begin{pmatrix}x-1\\n-1\end{pmatrix}}$, which is not possible… As their writing of $(7)$ seems to flow from the writing of $(3)$, it is that I did not understand anything about how to concretely use a prior for $N$ which is an improper uniform prior on all positive integers... Could you make me explicit how it's possible to go from formula $3$ to formula $7$?