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Math Help - Aternating Series Test

  1. #1
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    Aternating Series Test

    Hello,

    As a lemma to proving the "alternating series test" using the Cauchy criterion, I am told I need to show the following, (and then use this fact to prove the AST). (I am told it should be no more than a sentence or two)

    Lemma. Let a_n be a positive monotonically decreasing sequence. Suppose that for all N \in \mathbb{N}, m \geq n > N implies that \left| \sum_{k=n}^{m}(-1)^k a_k \right| \leq a_N . Then \sum_{k=0}^{\infty} (-1)^ka_k satisfies the Cauchy criterion

    (A series \sum a_n is said to satsify the cauchy crtierion if and only if for all \epsilon > 0, there exists N such that m \geq n >N imply \left|\sum_{k=n}^{m}a_n\right|< \epsilon

    NOTE: I am not asking for the actual proof of the AST. Just this basic fact above which is a prerequisite to the proof.

    Thanks for any help,

    James
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Aternating Series Test

    Do you know Leibniz convergence criterium?
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  3. #3
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    Re: Aternating Series Test

    No sir I do not. Again, my goal is not to prove the actual AST. I am just trying to justify a preliminary assumption. Perhaps this will make what I want a little clearer.


    Statement 1. For all N \in \mathbb{N}, m \geq n > N implies \left|\sum_{k=n}^m(-1)^ka_k\right| \leq a_N

    Statement 2. For all \epsilon > 0, there exists N \in \mathbb{N} such that m \geq n > N implies \left| \sum_{k=n}^m (-1)^k a_k \right| < \epsilon

    I need to show that Statement 1. implies Statement 2. (Prove that Statement 1. is sufficient for Statement 2.). My teacher explained that it should be very short.

    Thanks again,

    James
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