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Help with a convergence proof

  1. Aug 21, 2012 #1
    1. The problem statement, all variables and given/known data

    If xn-> ∞ then xn/xn+1 converges.

    2. Relevant equations



    3. The attempt at a solution

    I can see why the statement is true intuitively, but do not know how to make a rigorous proof. I have looked at the definitions of divergence/convergence but can get any ideas of how to prove this. Do I maybe start by showing that xn/xn+1 is bounded?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 21, 2012 #2

    Dick

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    It's not true. Try to find a counterexample.
     
  4. Aug 21, 2012 #3
    I'm pretty sure it's true, since each successive term of the sequence will be larger or equal to the previous, so xn/xn+1 should always be ≤ 1
     
  5. Aug 21, 2012 #4

    Dick

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    That would be true if the convergence were monotone (i.e. xn is increasing). But even if it were, that wouldn't prove it converges. There are a lot of numbers between 0 and 1.
     
  6. Aug 21, 2012 #5
    Well would I be able to claim that it is non-decreasing monotone, and show that it is bounded which implies convergence?
     
  7. Aug 21, 2012 #6

    Dick

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    (1,1,2,2,4,4,8,8,16,16,...). Does it converge to infinity? What about your ratio?
     
  8. Aug 21, 2012 #7
    Ahh I'm so sorry haha. The instructor just emailed us that there was a typo and the ratio should actually be xn / (xn+1). This makes more sense now. Thanks for your help though
     
  9. Aug 21, 2012 #8

    Dick

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    No problem, you're welcome. The correction makes a BIG difference.
     
  10. Aug 22, 2012 #9
    I still can't come up with an answer and my presentation is at 10.

    So far I've been able to show that since xn→∞, then 1/xn→0. Then

    1/(xn+1) < 1/xn < ε

    If anyone is available to help me, that would be very appreciated.
     
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