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Sequence Convergence proof

  1. Mar 14, 2013 #1
    1. The problem statement, all variables and given/known data
    Let {a_n} be a sequence | (a_n+1)^2 < (a_n)^2, 0 < (a_n+1) + (a_n). Show that the sequence is convergent


    2. Relevant equations

    n/a

    3. The attempt at a solution

    So I am feeling like monotone convergence theorem is the way to go there. It seems to me that (a_n+1)^2 < (a_n)^2 would imply the sequence is decreasing, but I do not know what to do with 0 < (a_n+1) + (a_n) to show it is bounded.
     
  2. jcsd
  3. Mar 14, 2013 #2

    mfb

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    Without the second inequality, you could construct series like 1+1/2, -1-1/3, 1+1/4, -1-1/5, ... - it has to be bounded based on the first inequality alone, but this is not sufficient for convergence.
    With both inequalities, you can rule out sign switches of a_n and get monotony.
     
  4. Mar 14, 2013 #3
    I do not understand how the first inequality shows that a_n is bounded.
     
  5. Mar 15, 2013 #4

    mfb

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    ##a_{n+1}^2 < a_n^2## is equivalent to ##|a_{n+1}| < |a_n|##, which leads to ##|a_{n}| < |a_0|\, \forall n \in \mathbb{N}##.
     
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