I am trying to find the maximum of the following function (mapping $[0,1]\rightarrow\mathbb{R}_{\ge0})$:
$$\max_{\alpha\in[0,1]} \frac{\sum_{i=1}^n i^2~ \prod_{j=i}^n \frac{j+1}{\lceil{\alpha j}\rceil}}{\sum_{i=1}^n \lceil\alpha i\rceil^2~ \prod_{j=i}^n \frac{j+1}{\lceil{\alpha j}\rceil} + \prod_{j=1}^{n}\frac{j+1}{\lceil{\alpha j}\rceil}},$$
ideally in the limit when $n\rightarrow +\infty$. This number is the best achievable price-of-anarchy in affine congestion games and was first introduced here (page 89) in a different context. As $n$ grows large, this number approaches $$2.01206694843168.$$
Q: Finding this number in closed form is very dear to me. Any suggestions/ideas on how to proceed?
Edit: Here is a plot of the function for $n=60$ (discretization in $\alpha$ is $10^{-6}$)
