Find a polynomial approximation for the maximum of $(y-y^n)/(2-y)$ I am trying to find the maximum value of  $$\frac{y-y^n}{2-y}$$ for $1>y>0$. I got that the maximum point should satisfy
$$(n-1)\cdot y^n-2ny^{n-1}+2=0.$$
and I know that the maximum should be $y=1-f(n)$, where $f(n)\to 0$ as $n$ approaches infinity.
I wish to find a polynomial approximation for $f(n)$, if one exists. Of course, $f(n)$ should be to polynomial by either $\frac{1}{n}$ or by $\frac{\ln n}{n}$.
 A: In fact, what you are asking is : what is the inverse function of
$$n=-\frac y{2-y}+\frac {W(t)}{\log(y)} \qquad \text{where}\qquad t=\frac 2{2-y} y^{\frac 2{2-y}} \log(y)$$ where $W(t)$ is Lambert function.
As you can suspect, this seems to be impossible and only numerical methods could be used.
In terms of polynomials, it seems that we need some high degree expansions. Detailed analysis shows that the best form would be
$$f(n)=\sum_{m=1}^p \frac{a_m} {\left(n^k\right)^m}$$ where $k <1$ which, more than likely, hides some logarithmic contibution.
I give you below $f(n)$ obtained using a nonlinear regression (the coefficients were made rational); the range used is $2 \leq n \leq 5000$. Using $k=\frac{2146}{2533}$, the coefficients are
$$\left\{\frac{1701}{776},-\frac{12793}{900},\frac{142354}{1305},-\frac{749471}{1340}
   ,\frac{2272141}{1389},-\frac{1174071}{479},\frac{402555}{278}\right\}$$
The maximum absolute error is $<0.00007$ and $R^2=0.999994$.
All coefficients are highly significant.
