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Math Help - determine convergence, absolute or conditional

  1. #1
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    determine convergence, absolute or conditional

    Suppose that \displaystyle \sum a_n and \displaystyle \sum b_n are absolutely convergent series and \displaystyle \sum c_n is conditionally convergent. For each of the series \displaystyle \sum(a_n+b_n), \displaystyle \sum(a_n+c_n) and \displaystyle \sum (a_nc_n), determine whether is it absolutely convergent or conditionally convergent or neither.

    I have already given the answer which are absolutely,conditionally and absolutely, I just need some explaination for \displaystyle \sum(a_n+c_n) and \displaystyle \sum (a_nc_n), why is conditionally and absolutely
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  2. #2
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    Quote Originally Posted by wopashui View Post
    Suppose that \displaystyle \sum a_n and \displaystyle \sum b_n are absolutely convergent series and \displaystyle \sum c_n is conditionally convergent. For each of the series \displaystyle \sum(a_n+b_n), \displaystyle \sum(a_n+c_n) and \displaystyle \sum (a_nc_n), determine whether is it absolutely convergent or conditionally convergent or neither.
    I have already given the answer which are absolutely,conditionally and absolutely, I just need some explaination for \displaystyle \sum(a_n+c_n) and \displaystyle \sum (a_nc_n), why is conditionally and absolutely
    For \displaystyle \sum(a_n+c_n) looking at the partial sums each series, A_n~\&~C_n.
    We can rearrange finite sums without effecting the outcome.
    Can you give an example that is not absolutely convergent?

    Because \displaystyle \sum c_n is conditionally convergent does that not imply that c_n in bounded?
    Say that B>0 and |c_n|\le B what can you say about |a_nc_n|~?
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  3. #3
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    Quote Originally Posted by Plato View Post
    For \displaystyle \sum(a_n+c_n) looking at the partial sums each series, A_n~\&~C_n.
    We can rearrange finite sums without effecting the outcome.
    Can you give an example that is not absolutely convergent?

    Because \displaystyle \sum c_n is conditionally convergent does that not imply that c_n in bounded?
    Say that B>0 and |c_n|\le B what can you say about |a_nc_n|~?

    so  |a_nc_n| <=B|a_n|, and by the comparion test, we know B|a_n| converges absolutely, thus |a_nc_n| converges absolutely. Right?

    Does conditional and absolute convergent both imply the series is bounded?
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  4. #4
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    Quote Originally Posted by wopashui View Post
    so[tex]Does conditional and absolute convergent both imply the series is bounded?
    If \sum {a_n } converges then (a_n)\to 0.
    Any convergent sequence is bounded.
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