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Thread: Summon the Heroes-1

  1. November 28th 2009, 04:49 AM #1
    Xingyuan
    Xingyuan is offline
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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

    Problem 1

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

    show that:

    \lim_{n \rightarrow \infty}\frac{a_{n}}{(1+a_{1})(1+a_{2})...(1+a_{n}) }=0
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  2. November 28th 2009, 05:20 AM #2
    Laurent
    Laurent is offline
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    Nice one!

    Spoiler:

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

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

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


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

    \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. November 28th 2009, 08:25 AM #3
    Xingyuan
    Xingyuan is offline
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    Thumbs up

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