I am interested in the unique solution $x$ for the equation : $$ p_K(x)=\frac{x+\dots+x^K}{K}=\frac{1}{2}, $$ for large values of $K$. When $K$ is small ($K=1$ and $K=2$) we can solve this equation explicitly and find : $$ x=\frac{1}{2}, \frac{\sqrt{5}-1}{2}. $$ For $K=3$ we still get an explicit solution which is more complicated and from $K=4$ […]

- Tags \frac{\sqrt{5}-1}{2}. $$ For $K=3$ we still get an explicit solution which is more complicated and from $K=4$ on I do not find an explicit so, $$ for large values of $K$. When $K$ is small ($K=1$ and $K=2$) we can solve this equation explicitly and find : $$ x=\frac{1}{2}, find : $$ \lim_{K\rightarrow \infty} K \cdot \log(x_K) \approx - 1.592 $$, I am interested in the unique solution $x$ for the equation : $$ p_K(x)=\frac{x+\dots+x^K}{K}=\frac{1}{2}, i.e. $\lim_{K \rightarrow \infty} x_K = 1$. I would like to find an asymptotic approximation of $p_K(x)$ denoted by $\tilde p_K(x)$ for which, we find $\tilde x_K = \left( \frac{1}{2} \right) ^{\frac{2}{K+1}}$ and we find for the limit : $$ \lim_{K\rightarrow \infty} K \cdot \log(\ti, we find that the unique solution $x_K$ of $p_K(x_K)=0$ tends to one