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Homework Help: About Convergence of a particular sequence

  1. Sep 19, 2010 #1
    1. The problem statement, all variables and given/known data


    let's define {b_n} as an infinite sequence such that
    all terms are nonnegative, and
    for any i, j >= 1,

    b_{i+j} <= b_i + b_j

    it's easy to show that b_n/n <= (b_1) for all n, but
    how would you show that
    {(b_n)/n} is a convergent sequence?


    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution

    Using induction, I got upto 0 < = b_n / n < = b_1
    but that was pretty much it.
    I have to show that it's monotone from some N on or something like that, i think..
    but I'm just stuck about how to show that.
     
    Last edited: Sep 19, 2010
  2. jcsd
  3. Sep 19, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If you've shown b_n<b_1/n then b_n/n<b_1/n^2. Can you show b_1/n^2 is a convergent sequence? An integral test should work.
     
  4. Sep 19, 2010 #3
    my bad, problem fixed. i meant b_n < = b_1
     
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