# Sequences defined as solutions to equations : $u_{n}=v_{n}^n$ [closed]

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.

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Well, this is not a forum, but a Question & Answer site - please consider the faq. Moreover, this question is crossposted (math.stackexchange.com/questions/85045/…) which is considered bas style since it wastes time and may cause confusion. –  Dirk Nov 23 '11 at 21:15
Voting to close. If you don't get anything at m.se after a week or so, try here again (but do link to the m.se discussion). –  Gerry Myerson Nov 23 '11 at 21:36
It doesn't seem to be even close to true for small values of n. I don't know why this question is here. –  Felipe Voloch Nov 23 '11 at 22:54
$u_2=0.29559774\dots$ is a zero of $x^3+x^2+3x-1$ and $v_2=(\sqrt5-1)/2$ is a zero of $x^2+x-1$. –  Wadim Zudilin Nov 23 '11 at 23:22