We consider the infinite product:


$$\frac{1}{c}\prod\_{n \geqslant 1} \frac{c^n}{c^n+1} = \frac{1}{c}\frac{c}{c+1}\frac{c^2}{c^2+1}\frac{c^3}{c^3+1} \cdots$$


For which real value of $c > 1$ has the product its maixmum?


Background of Question: Expected number of equilibria in multi-person games on trees of uniform width c, c=3 gives maximum fraction of equilibria for integral c. Allowing real-valued c may give more insight.


Ingo.



Maximise specific infinite product
Optimal $c$ is the unique positive root of $\sum k/(1+c^k)=1$, numerically about 2.43. I doubt that it is expressable via known constants and elementary functions.


