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Prove a_n is divergent?

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose that ∑a_n and ∑b_n are series with positive terms and ∑b_n is divergent. Prove that if:

    lim a_n/b_n = infinity
    n--->infinity

    then ∑a_n is also divergent.


    2. Relevant equations



    3. The attempt at a solution

    Well in attempting to write a viable solution, I have deducted that since both series have positive terms, both sequences are increasing. If ∑b_n is is divergent and the limit as n approaches infinity of a_n/b_n is infinity than ∑a_n also must be divergent. Is there anymore to this however? I think I am missing something important in the explanation but I am not too sure of what it is. Thank you!
     
  2. jcsd
  3. Mar 19, 2012 #2

    lanedance

    User Avatar
    Homework Helper

    I would start with your definition of divergnence, what is it?

    Qualitatively, hopefully you can see what is going on the series bn diverges, but for some n>N, every term is an is much larger that the bn term hence the sum over an diverges

    an example is:
    [tex] b_n = \frac{1}{n}[/tex]
    [tex] a_n = \frac{1}{\sqrt{n}}[/tex]
     
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