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How to show sequnce is divergent (math. induction)

  1. Oct 3, 2012 #1
    1. The problem statement, all variables and given/known data

    For the sequnce an defined recursively, I have to determine the limiting value, provided that it exists.
    a1=2 and a(n+1) = 1/(an)^2 for all n

    2. Relevant equations



    3. The attempt at a solution

    Ok, so I 've done problems like this but the a(n+1) was biggger than an and so the sequence was increasing and I used mathematical induction and boundness to show it converged. Then I founf the limit.
    Now, though I'm having trouble with this sequence becasue it's a decreasing sequence and I know it diverges. How do I go about proving that? What method do I use? At what point is it realized that it diverges?
    I'm just really lost :/

    Thank you!
     
  2. jcsd
  3. Oct 3, 2012 #2

    SammyS

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    Have you written out the first several terms of the sequence?
     
  4. Oct 3, 2012 #3
    I have....1/4, 1/16...etc
     
  5. Oct 3, 2012 #4

    Mark44

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    a1 = 2
    a2 = 1/(22) = 1/4
    a3 = ?
    Hint: It's not 1/16.
     
  6. Oct 3, 2012 #5
    Isn't it 1/(2^2)^2
     
  7. Oct 3, 2012 #6

    Mark44

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    Not according to what you wrote in post 1.


     
  8. Oct 3, 2012 #7

    SammyS

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    "It", as in a3 is

    [itex]\displaystyle a_3=\frac{1}{{a_2}^2}[/itex]
    [itex]\displaystyle
    =\frac{1}{(1/4)^2}[/itex]

    [itex]\displaystyle =\frac{1}{1/16}[/itex]

    [itex]= \underline{\ \ ?\ \ }[/itex]
     
  9. Oct 3, 2012 #8

    Mark44

    Staff: Mentor

    You shouldn't say "it" unless it is crystal clear what the antecedent is.
     
  10. Oct 3, 2012 #9
    Oh wow, stupid mistake. Thanks!
    So, do I need to show anything to show divergence or just state that it's going to zero and infinity..oscillating, so it's divergent?
     
  11. Oct 3, 2012 #10
    Well there certainly are two subsequences that converge to 0 and diverge to infinity.
     
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