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Math Help - Analysis: alternating series

  1. #1
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    Analysis: alternating series

    Let (a_n)_{n\in\mathbb{N}} be a non-increasing non-negative sequence.

    Prove that s_n= \sum_{i=1}^{n}{(-1)^{i+1}a_i} is bounded and that \limsup{s_n}-\liminf{s_n}=\lim{a_n}

    I would appreciate any help.
    Last edited by JoachimAgrell; January 28th 2010 at 07:13 PM.
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  2. #2
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    Quote Originally Posted by JoachimAgrell View Post
    Let (a_n)_{n\in\mathbb{N}} be a non-increasing non-negative sequence.

    Prove that s_n= \sum_{i=1}^{n}{(-1)^{n+1}a_i} is bounded and that \limsup{s_n}-\liminf{s_n}=\lim{a_n}

    I would appreciate any help.
    Maybe I'm reading this wrong but what about a_i=1, it's clearly non-increasing and non-negative but s_n=(-1)^{n+1}n which is clearly not bounded
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  3. #3
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    Oops. I meant (-1)^{i+1} in that series.
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  4. #4
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    Quote Originally Posted by JoachimAgrell View Post
    Let (a_n)_{n\in\mathbb{N}} be a non-increasing non-negative sequence.

    Prove that s_n= \sum_{i=1}^{n}{(-1)^{i+1}a_i} is bounded and that \limsup{s_n}-\liminf{s_n}=\lim{a_n}

    I would appreciate any help.
    Okay here's my attempt:

    By induction we prove that s_n\leq a_1 and thus we get that s_n is bounded because it's non negative. Now it's clear that s_{2n-1} \geq s_{2n} for all n and since each of these are monotonic we get that s_n has at most two acc. points and so s_{2n-1} \rightarrow \limsup s_n and s_{2n} \rightarrow \liminf s_n but s_{2n-1} -s_{2n}= a_{2n} \rightarrow \lim a_n
    Last edited by Jose27; January 28th 2010 at 09:19 PM.
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