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Thread: prove that (arctan x)<x if x is positive

  1. #1
    Senior Member
    Feb 2008

    prove that (arctan x)<x if x is positive

    It seems like this should be a fairly simple thing, yet I can't seem to pin down the trick.

    Hi, all. I need to prove that if $\displaystyle x>0$, then $\displaystyle \tan^{-1}x<x$.

    Any ideas?

    POTENTIALLY USELESS INFO: This is in order to prove that $\displaystyle \{x_n\}$ recursively defined by $\displaystyle x_{n+1}=\tan^{-1}x_n$ $\displaystyle \forall$ $\displaystyle n\geq 0$ and $\displaystyle x_0>0$ converges to a finite limit.

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  2. #2
    Junior Member
    Aug 2009
    x - arctan(x) is continuous and is 0 at 0, its derivative is 1 - 1/(1+x^2) which is positive when x>0 .... I see no problem here, it is an increasing function.
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