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Finding an oscillating sequence that diverges and whose limit is zero.

  1. Mar 10, 2008 #1
    1. The problem statement, all variables and given/known data

    Hi, I need to find an oscillating sequence whose limit of the differences as n approaches infinity is zero but the sequence itself is diverging.

    2. Relevant equations

    None.

    3. The attempt at a solution

    My initial guess was:

    [tex]\frac{sin(ln(n))}{ln(n)}[/tex]
     
  2. jcsd
  3. Mar 10, 2008 #2

    HallsofIvy

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    What do you mean by an "oscillating" sequence?
     
  4. Oct 5, 2009 #3
    Hi!

    If I understand correctly, then
    [tex]sin\left(\sqrt{n}\right)[/tex]
    is what you are looking for. The proof is based on the idea sin(a)-sin(b) < a-b and sqrt(n+1)-sqrt(n) converges to 0.
    I originally bumped this post, because I need help with a similar problem, where not
    an+1-an converges to 0, but an+1/an converges to 1.
    Does anybody know a good example for the latter? So the problem again:
    We need a sequence, that:
    1) oscillates (oscillation is when it divergates, but neither to infinity nor negative infinity, eg. -1 +1 -1 +1 ... ; +1 -2 +3 -4 +5 -6 ... ;sin(n) ; etc.)
    2) an+1/an converges to 1
     
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