I got the objective function $\displaystyle f(\alpha)=\alpha\cdot \left(1-\frac{\binom{N+K}{K}\beta^K}{\sum\limits_{k=0}^{K}\binom{N+k}{k}\beta^k}\right)$, where $N$ and $K$ are positive integers, $\displaystyle\beta=\frac{\alpha}{1-\alpha}$. Then, how to solve or deal with the following optimization issue
\begin{equation} \max\limits_{\alpha\in[0,1]}f(\alpha) \end{equation}
This is not an answer but I hope it could give you some ideas.
$$A=\sum\limits_{k=0}^{K}\binom{N+k}{k}\beta^k=(1-\beta )^{-N-1} (1-(K+1) \binom{K+N+1}{K+1} B_{\beta }(K+1,N+1))$$ where appears the Beta function.
It is then possible to simplify $$\frac{\binom{N+K}{K}\beta^K}{\sum\limits_{k=0}^{K}\binom{N+k}{k}\beta^k}=\frac{ \beta ^K (1-\beta )^{N+1} (K+N)!}{K! N!-(K+N+1)! B_{\beta }(K+1,N+1)}$$
Considering the function, we have $f(0)=0$, $f(1)=0$; so, for a starting guess, I should compute the intersection of the tangents at the end points. For the optimization, I suppose that there will not be any particular problem.