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Thread: Summon the Heroes-1

  1. #1
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    Thumbs up Summon the Heroes-1

    From now on, I will post some problem to play with,If this problem be worked out, The next will be posted.

    Let's Enjoy the great game...

    Have Fun


    $\displaystyle Problem$ $\displaystyle 1$

    If $\displaystyle a_{n}>0$ for $\displaystyle \forall$ $\displaystyle n \in \mathbb{N}^+$

    show that:

    $\displaystyle \lim_{n \rightarrow \infty}\frac{a_{n}}{(1+a_{1})(1+a_{2})...(1+a_{n}) }=0$
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  2. #2
    MHF Contributor

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    Paris, France
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    Nice one!

    Spoiler:

    The sequence of general term $\displaystyle (1+a_1)\cdots (1+a_n)$, $\displaystyle n\geq 1$, is increasing, therefore either it has a finite limit, or it diverges to $\displaystyle +\infty$.

    If it has a finite limit $\displaystyle \ell$ (note that $\displaystyle 1\leq\ell$), then $\displaystyle a_n\to 0$ because the ratio $\displaystyle 1+a_n$ of two consecutive terms in the previous sequence tends to $\displaystyle \frac{\ell}{\ell}=1$. Therefore,

    $\displaystyle \frac{a_n}{(1+a_1)\cdots(1+a_n)}\leq a_n\to 0.$


    And if it diverges to $\displaystyle +\infty$, then, using $\displaystyle \frac{a_n}{1+a_n}\leq 1$, we have

    $\displaystyle \frac{a_n}{(1+a_1)\cdots(1+a_n)}\leq \frac{1}{(1+a_1)\cdots(1+a_{n-1})}\to 0$
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  3. #3
    Junior Member
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    Thumbs up

    Elegant and powerful !!!
    Last edited by Xingyuan; Nov 28th 2009 at 08:37 AM.
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