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Thread: Use the formal definition to prove that the following sequence diverges

  1. #1
    MHF Contributor alexmahone's Avatar
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    Use the formal definition to prove that the following sequence diverges

    $\displaystyle s_n=\left(\frac1n-1\right)^n$


    My attempt:


    For large $n$, the sequence oscillates between $e^{-1}$ and $-e^{-1}$ and therefore diverges. Now for the proof.


    Assume, for the sake of argument, that the sequence converges to $L$.


    $\exists N\in\mathbb{N}$ such that $|s_n-L|<0.1$ whenever $n\ge N$


    $\displaystyle\left|\left(\frac1n-1\right)^n-L\right|<0.1$ whenever $n\ge N$


    $\displaystyle\implies\left|\left(\frac1{n+1}-1\right)^{n+1}-L\right|<0.1$ whenever $n\ge N$


    We can rewrite these 2 equations as


    $\displaystyle\left|(-1)^n\left(1-\frac1n\right)^n-L\right|<0.1$ whenever $n\ge N$ --------------- (1)


    $\displaystyle\left|(-1)^{n+1}\left(1-\frac1{n+1}\right)^{n+1}-L\right|<0.1$ whenever $n\ge N$


    $\displaystyle\implies\left|(-1)^n\left(1-\frac1{n+1}\right)^{n+1}+L\right|<0.1$ whenever $n\ge N$ --------------- (2)


    How do I get a contradiction from equations (1) and (2)?
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  2. #2
    MHF Contributor

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    Re: Use the formal definition to prove that the following sequence diverges

    You don't. It is quite possible that there exist n and n+ 1 such that they are both within 0.1 of L. You need to show that there exist n such that this is not within any \epsilon> 0.
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