MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

closed as off topic by Felipe Voloch, Gerry Myerson, Pietro Majer, Wadim Zudilin, Douglas Zare Nov 24 '11 at 0:05

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Well, this is not a forum, but a Question & Answer site - please consider the faq. Moreover, this question is crossposted (…) 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 after a week or so, try here again (but do link to the 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