Results 1 to 4 of 4

Thread: determine convergence, absolute or conditional

  1. #1
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    determine convergence, absolute or conditional

    Suppose that $\displaystyle \displaystyle \sum a_n$ and $\displaystyle \displaystyle \sum b_n $are absolutely convergent series and $\displaystyle \displaystyle \sum c_n $is conditionally convergent. For each of the series $\displaystyle \displaystyle \sum(a_n+b_n)$, $\displaystyle \displaystyle \sum(a_n+c_n) $and $\displaystyle \displaystyle \sum (a_nc_n)$, 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 \displaystyle \sum(a_n+c_n) $and $\displaystyle \displaystyle \sum (a_nc_n)$, why is conditionally and absolutely
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,776
    Thanks
    2823
    Awards
    1
    Quote Originally Posted by wopashui View Post
    Suppose that $\displaystyle \displaystyle \sum a_n$ and $\displaystyle \displaystyle \sum b_n $are absolutely convergent series and $\displaystyle \displaystyle \sum c_n $ is conditionally convergent. For each of the series $\displaystyle \displaystyle \sum(a_n+b_n)$, $\displaystyle \displaystyle \sum(a_n+c_n) $and $\displaystyle \displaystyle \sum (a_nc_n)$, 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 \displaystyle \sum(a_n+c_n) $and $\displaystyle \displaystyle \sum (a_nc_n)$, why is conditionally and absolutely
    For $\displaystyle \displaystyle \sum(a_n+c_n) $ looking at the partial sums each series, $\displaystyle A_n~\&~C_n$.
    We can rearrange finite sums without effecting the outcome.
    Can you give an example that is not absolutely convergent?

    Because $\displaystyle \displaystyle \sum c_n $ is conditionally convergent does that not imply that $\displaystyle c_n$ in bounded?
    Say that $\displaystyle B>0$ and $\displaystyle |c_n|\le B$ what can you say about $\displaystyle |a_nc_n|~?$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2010
    Posts
    273
    Quote Originally Posted by Plato View Post
    For $\displaystyle \displaystyle \sum(a_n+c_n) $ looking at the partial sums each series, $\displaystyle A_n~\&~C_n$.
    We can rearrange finite sums without effecting the outcome.
    Can you give an example that is not absolutely convergent?

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

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

    Does conditional and absolute convergent both imply the series is bounded?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,776
    Thanks
    2823
    Awards
    1
    Quote Originally Posted by wopashui View Post
    so[tex]Does conditional and absolute convergent both imply the series is bounded?
    If $\displaystyle \sum {a_n } $ converges then $\displaystyle (a_n)\to 0$.
    Any convergent sequence is bounded.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Conditional and absolute convergence
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 4th 2010, 03:07 PM
  2. Absolute or conditional convergence
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Mar 31st 2009, 08:15 PM
  3. Replies: 6
    Last Post: Mar 15th 2009, 07:28 PM
  4. Absolute and conditional convergence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Jan 25th 2009, 02:57 PM
  5. absolute or conditional convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 16th 2008, 11:29 AM

Search Tags


/mathhelpforum @mathhelpforum