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Math Help - Alternating Series Test

  1. #1
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    Alternating Series Test

    Prove the "Alternating Series Test"; .ie suppose that (x_n) is a positive
    decreasing sequence with lim (x_n)= 0. Show that the alternating series


    (-1)^n (x_n)
    n=0

    is convergent and the sum satisfies |s-s_k|≤ s_k

    where s_k is the partial sum and s = lim (s_k).

    Hint: Start by showing that (s_2k) and (s_2k+1) are both monotone.



    --------------


    I showed that the odd partial sums are monotone increasing and the even partial sums are monotone decreasing, but I couldn't find the rest.

    If you help me, I will be really happy.

    Selin
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  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
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    Hello,
    Quote Originally Posted by selinunan View Post
    Prove the "Alternating Series Test"; .ie suppose that (x_n) is a positive
    decreasing sequence with lim (x_n)= 0. Show that the alternating series


    (-1)^n (x_n)
    n=0

    is convergent and the sum satisfies |s-s_k|≤ s_k
    Don't you mean |s-s_k|\leq x_{k+1} ?

    I showed that the odd partial sums are monotone increasing and the even partial sums are monotone decreasing, but I couldn't find the rest.
    Can you show that (s_{2k}) is bounded from below and that (s_{2k+1}) is bounded from above ? Hint (highlight to read) : * show that for all k, s_{2k+1}<s_{2k} *
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  3. #3
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    In the question it is written as the way I wrote:

    |s-s_k|≤ s_k

    But I think it should have be written as you have said.

    It can't be solved if it's |s-s_k|≤ s_k , can it?
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