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Thread: Limit 3

  1. #1
    Junior Member
    Joined
    Dec 2009
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    Limit 3

    Let $\displaystyle f$ be function for which which holds following conditions

    $\displaystyle F\in C^1(R)\mbox{ and }|f'(x)|<1, \forall x\in R$
    $\displaystyle f(x+1)=f(x)\ ,\forall x\in R, f \mbox{ is periodical, with } \omega=1)$

    Lets define function $\displaystyle p:R\rightarrow R$ with $\displaystyle p(x)=x+f(x)$.
    Prove that
    $\displaystyle \displaystyle
    \lim\limits_{n\to\infty}\frac{{x+p(x)+p(p(x))+\dot s + \overbrace{p(p(\dots (p(x))\dots))}^{(n-1)-\text{times}}}}{{n^2}}
    $

    exist, and doesn't depends on $\displaystyle x$.
    Last edited by ns1954; Jan 21st 2010 at 10:12 AM.
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