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Math Help - Proving Divergence

  1. #1
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    Proving Divergence

    The definition of a convergent squence is:
    For every L in Real numbers, and all E>0 such that there exists N in Natural numbers n>/= N implies |xn-L|>E.
    Negate this statement then use the negation to prove that the sequence xn=(-1)^n is not convergent.

    I got the negation I think, but I'm not sure how to begin with the proof. We went over one example in class and it was for proving something converges. I got the negation as :
    For all L in real numbers, there exists E>0 such that there exits N in natural numbers and n>/=N implies |xn-L|<E
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    Re: Proving divergence

    Your definition of convergence is not correct. It should be

    There exists L in Real numbers such that for all E>0 such that there exists N in Natural numbers such that n>/= N implies |xn-L|< E.

    The negation of the definition of convergence should be: For every real number L there exists a real number \varepsilon>0 such that for every natural number N, there is a natural number n\geqslant N such that \left|x_n-L\right|\geqslant\varepsilon. Use this to prove the divergence of \left(x_n\right)_{n=1}^\infty where x_n=(-1)^n. Hint: Given L, let \varepsilon=\min\left\{|L+1|,|L-1|\right\} if L\ne\pm1 and \varepsilon=2 if L=\pm1.
    Last edited by Sylvia104; March 30th 2012 at 12:47 PM.
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    Re: Proving Divergence

    I don't understand how you are proving the negation. We just used the pieces of the definition to prove convergence in class. Is that what you are doing here?
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    Re: Proving Divergence

    Quote Originally Posted by renolovexoxo View Post
    I don't understand how you are proving the negation. We just used the pieces of the definition to prove convergence in class. Is that what you are doing here?

    Is this true: \left( {\forall n} \right)\left[ {\left| {{x_{n + 1}} - {x_n}} \right| = 2} \right]~?
    If that is true, how can the terms of the sequence {\left( { - 1} \right)^n} get "close together"?
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    Re: Proving Divergence

    I'm getting lost. I feel like there is something there that I'm not understanding.
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    Re: Proving Divergence

    Quote Originally Posted by renolovexoxo View Post
    I'm getting lost. I feel like there is something there that I'm not understanding.
    I think that you need a live-sit-down with an instructor/lecturer.
    We here just not equipped to aid with this kind profound confusion.
    Talk to your instructor!
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  7. #7
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    Re: Proving Divergence

    I can't or I would. Thanks for trying I suppose. I understand the second piece, but between the first response and the first line of what you wrote I can't figure it out. Do I only need the fact that they cannot equal 2 or get closer together for all n? That's really where I'm getting hung up. If asked to explain it, I could, it's more where I am supposed to get this fact from that is causing an issue. Should I just have been able to know that?
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