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Math Help - Proof of alternating series test

  1. #1
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    Proof of alternating series test

    Hi,

    I'm having trouble understanding the first line of Knapp's proof of the alternating series test. The Theorem states that

    If for each x in a nonempty set S, \{ a_n(x)\}_{n \geq 1} is a monotone decreasing sequence of nonnegative real numbers such that \lim_n a_n(x) = 0 uniformly in x, then \sum^{\infty}_{n=1}(-1)^n a_n(x) converges uniformly.

    His opening line in the proof is

    The hypotheses are such that \left|  \sum^{n}_{k=m}(-1)^k a_k(x) \right| \leq \sup_x |a_m(x)| whenever n \geq m...

    It's the above line I'm totally confused by. Can someone help explain to me how the hypothesis implies the above?

    Thanks
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  2. #2
    Super Member girdav's Avatar
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    I won't give the complete details but the idea is the following. We have
    0\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}\leq a_m(x) and \leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq a_m(x).
    Last edited by girdav; June 9th 2011 at 01:38 PM.
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  3. #3
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    Quote Originally Posted by girdav View Post
    I won't give the complete details but the idea is the following. We have
    0\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}\leq a_m(x) and a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq 0.
    If a_n(x) is nonnegative and monotone decreasing, how is a_m(x)-a_{m+3}(x)\leq 0?
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  4. #4
    Super Member girdav's Avatar
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    Sorry, the second inequality is wrong, I meant a_m(x) instead of 0. I will correct it.
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  5. #5
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    Ok, I think I see it now; I'll work on writing the argument out in full. Thanks
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