Results 1 to 4 of 4
Like Tree2Thanks
  • 1 Post By Plato
  • 1 Post By Plato

Math Help - Subsequence Convergence

  1. #1
    Member
    Joined
    Mar 2012
    From
    USA
    Posts
    92

    Subsequence Convergence

    Suppose ak>= 0 for all k in N. Prove that the series from 1 to inifity of ak converges iff some subsequence {snk} of the sequence {sn} of partial sums converges.

    I've proved that if ak converges then the subsequence will converge. I'm have some trouble going the reverse dirrection, proving that if the subsequence converges then ak will converge. Can I just take {snk}={sn} and then by our assumption, since the sequence of partial sums converges the series ak will converge?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1

    Re: Subsequence Convergence

    Quote Originally Posted by renolovexoxo View Post
    Suppose ak>= 0 for all k in N. Prove that the series from 1 to inifity of ak converges iff some subsequence {snk} of the sequence {sn} of partial sums converges.
    Because a_n\ge 0 then the sequence of partial sums is increasing.

    Now suppose that \left( {S_{n_i } } \right) \to S.

    If \varepsilon  > 0 then \left( {\exists K} \right)\left[ {S - \varepsilon  < S_{n_K }  \leqslant S} \right].

    But if N\ge n_K you know that  {S - \varepsilon  < S_N  \leqslant S} .
    Thanks from renolovexoxo
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2012
    From
    USA
    Posts
    92

    Re: Subsequence Convergence

    Would you mind explaining the very last inequality just a little further for me? I follow you until that conclusion, then I'm a little lost.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1

    Re: Subsequence Convergence

    Quote Originally Posted by renolovexoxo View Post
    Would you mind explaining the very last inequality just a little further for me? I follow you until that conclusion, then I'm a little lost.
    In effect, it shows that the sequence of partial sums converges: \left( {S_n } \right) \to S.

    Again that is because the sequence of partial sums is increasing.
    Thanks from renolovexoxo
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subsequence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 5th 2010, 05:17 AM
  2. subsequence
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 25th 2009, 01:27 AM
  3. subsequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 15th 2009, 11:17 AM
  4. convergence of sequence and subsequence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 8th 2008, 12:20 PM
  5. Subsequence
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 24th 2006, 09:43 AM

Search Tags


/mathhelpforum @mathhelpforum