Hello, I hope this is the appropriate forum for posting. For $n$ a positive integer at least equal to $2$, define the two following functions as follows : $f_{n}(x)=\pi/4 + \arctan(\sqrt[n]{x})-\arctan(1/x)$ for all nonzero $x$. $g_{n}(x)=x^{n+1} +x^n+x-1$ for all real $x$. We can find, using differentiation for instance, that these two functions have each one unique real root and, using the intermediate value theorem, that it lies in the interval $(0,1)$. If we define the sequence $u_{n}$ as the real root of $f_{n}$ and $v_{n}$ as the real root of $g_{n}$, my question is to prove that the equality $u_{n}=v_{n}^n$ holds for all $n\geq 2$.
EDIT : one should read $g_{n}(x)=x^{n+1} +x^n+x-1$ instead of $g_{n}(x)=x^{n+1} +x^4+x-1$. Now corrected.
