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Converrgence of oscillating sequence

  1. Aug 6, 2004 #1
    Convergence of oscillating sequence

    Hi, I have to prove that an oscillating sequence converges, I am having some difficulty with the proof.

    The sequence is [tex] c_{n+1} = \frac{1}{1+c_{n}} , c_{1} = 1[/tex]

    So, I've calculated the first few terms and have seen that the sequence oscillates. I know that I need to prove:
    1) The differences alternate in sign.
    2) The absolute differences decrease.
    3) The absolute differences approach 0.

    I have proved 1, using:

    [tex]
    c_{n+1} - c_{n} = \left(\frac{1}{1+c_{n}}\right) - \left(\frac{1}{1+c_{n-1}}\right)
    =\frac{1+c_{n-1}-1-c_{n}}{1+c_{n-1}+c_{n}+c_{n-1}c_{n}}
    [/tex]
    [tex]
    =\frac{-(c_{n}-c_{n-1})}{1+c_{n-1}+c_{n}=c_{n-1}c_{n}}
    [/tex]

    And since all terms are positive, the denomenator will be positive and the difference between two terms with alternate in sign from the difference between the previous two terms.

    I now am having trouble proving 2 and 3. I'm not exactly sure what to do; the example in my book is not very helpful.So far I have:

    [tex]
    |c_{n+1}-c_{n}| < |c_{n} - c_{n-1}|
    [/tex]

    but that's not much... If anyone could help, that would be great!! Thanks!
     
    Last edited: Aug 6, 2004
  2. jcsd
  3. Aug 6, 2004 #2

    arildno

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    Note that:
    [tex](1+c_{n})(1+c_{n-1})=2+c_{n-1}[/tex]
    By substituting [tex]c_{n}=\frac{1}{1+c_{n-1}}[/tex]
     
  4. Aug 6, 2004 #3
    Thank you! Ok, now I have:

    [tex]
    |c_{n+1}-c_{n}| = \frac{|c_{n}-c_{n-1}|}{2+c_{n-1}}
    [/tex]

    And since all terms are positive, [tex]2+c_{n-1}[/tex] will be positive, and each absolute difference will be a fraction of the previous absolute difference. Therefore they are decreasing and they will approach 0 as n apporaches infinity. Is that enough to prove this by just saying this? Thanks!

    (Is there any way to change the title of the thread? I made a typo :redface: )
     
  5. Aug 6, 2004 #4

    arildno

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    Technically, I guess you should prove that the gained relations imply that we've got a Cauchy sequence, and hence, that the sequence converges (depends on what you may take as granted)
     
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