Results 1 to 6 of 6

Math Help - Series: Comparison test

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    london
    Posts
    8

    Series: Comparison test

    Suppose for an >= 0 for all n and \sum_{n=1}^\infty a_n is convergent. explain why an < 1 for all large enough n.
    Deduce that \sum_{n=1}^\infty \a_n^\2 is convergent.

    Could anyone lend a hand? I know you have to use the comparison test, but how?
    Thanks
    Last edited by algebra123; March 6th 2013 at 03:52 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,916
    Thanks
    1762
    Awards
    1

    Re: Series: Comparison test

    Quote Originally Posted by algebra123 View Post
    Suppose for an >= 0 for all n and \sum_{n=1}^\infty a_n is convergent. explain why an < 1 for all large enough n.
    Deduce that \sum_{n=1}^\infty \a_n^\2 is convergent.
    First, you must explain why it must be true that the sequence (a_n)\to 0.

    Once you have that (\exists N)[n\ge N\text{ implies }|a_n|<0.5]

    Now you know that the geometric series \sum\limits_{k = N}^\infty  {{{\left( {0.5} \right)}^k}} converges.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2012
    From
    london
    Posts
    8

    Re: Series: Comparison test

    I don't understand how you would know that (a_n) tends to 0 ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,916
    Thanks
    1762
    Awards
    1

    Re: Series: Comparison test

    Quote Originally Posted by algebra123 View Post
    I don't understand how you would know that (a_n) tends to 0 ?
    Then you have utterly failed to learn the first and perhaps most important theorem dealing with series convergence.

    Go look up the first test of divergence: If \sum\limits_{k = N}^\infty  {{a_k}} convergences only if what?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2012
    From
    london
    Posts
    8

    Re: Series: Comparison test

    Our lecturer is really far behind and hasn't been through any of this. I'm so confused, where did the 0.5 come from?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,916
    Thanks
    1762
    Awards
    1

    Re: Series: Comparison test

    Quote Originally Posted by algebra123 View Post
    Our lecturer is really far behind and hasn't been through any of this. I'm so confused, where did the 0.5 come from?
    I doubt we can help with what your lecturer has or has not done.
    That is a matter for you to take up with your educational authority.

    But you have a textbook/lecture notes. You can at least look up the what part of my question:
    \sum\limits_{k = N}^\infty  {{a_k}} convergences only if what?
    If you want more help, show some effort.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Comparison test and Limit Comparison test for series
    Posted in the Calculus Forum
    Replies: 5
    Last Post: November 25th 2010, 01:54 AM
  2. Series Comparison test
    Posted in the Calculus Forum
    Replies: 7
    Last Post: September 16th 2009, 05:45 AM
  3. Comparison Test: Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 19th 2009, 05:14 PM
  4. Limit comparison/comparison test series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 25th 2009, 09:27 PM
  5. Comparison & Limit Comparison test for series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 25th 2009, 05:00 PM

Search Tags


/mathhelpforum @mathhelpforum