Results 1 to 2 of 2

Math Help - monotone convergence problem

  1. #1
    Newbie
    Joined
    Oct 2007
    Posts
    2

    monotone convergence problem

    Let {bn} be a bounded sequence of nonnegative numbers and r be any number such that 0≤r<1.


    sn = b1r + b2r^2 + ….. + bnr^n for every index n.



    Use the monotone convergence theorem to prove that the series {sn} converges.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,386
    Thanks
    1476
    Awards
    1
    Because all these numbers are nonnegative, then it should be clear to you that \left( {s_n } \right) is an non-decreasing sequence. Now let B be the bound such that \left( {\forall n} \right)\left( {\left| {b_n } \right| \le B} \right).
    It should also be clear that, s_n  = \sum\limits_{k = 1}^n {b_k r^k }  \le B\sum\limits_{k = 1}^n {r^k }  \le \frac{{Br}}{{1 - r}}.

    What do you know about a bounded non-decreasing sequence?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Monotone
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: December 29th 2011, 04:30 AM
  2. Monotone convergence vs Dominated Convergence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 20th 2011, 09:11 AM
  3. Replies: 3
    Last Post: May 15th 2011, 02:13 AM
  4. analysis-bounded monotone sequence problem
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: April 24th 2010, 12:06 PM
  5. Replies: 1
    Last Post: September 26th 2007, 10:01 AM

Search Tags


/mathhelpforum @mathhelpforum