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Convergence of a series

  1. Aug 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Let [tex] x_{n}[/tex] be a monotone increasing sequence such that [tex] x_{n+1}-x_{n}< \frac{1}{n}.[/tex] Must [tex] x_{n}[/tex] converge ?


    2. Relevant equations

    Instinctively, I think it converges since the terms "bunch" up as n increases.

    3. The attempt at a solution
    [tex] |x_{n+1}-L| \leq |x_{n+1}-x_{n}| + |x_{n}-L|[/tex].
    [tex] |x_{n+1}-L|< \frac{1}{n}+ |x_{n}-L|[/tex] . But this doesn't tell me anything about convergence.

    If I apply the triangle inequality continuosly all I can see is that
    [tex] |x_{n+1}-L|< \sum\frac{1}{n}[/tex]

    What can you guys tell me ?
     
  2. jcsd
  3. Aug 16, 2010 #2
    Great that makes sense. I had a feeling it need not converge but I was unsure. Does this work too...
    Define a sequence such that [tex]x_{n}-x_{n-1}= \frac{1}{n}[/tex] such that the sequence has positive terms them
    [tex] |x_{n} -L| =|1 + 1/2 + ...+ 1/n|[/tex ] which does not converge.
    I suppose that also works correct?
    Excuse my latex, typing from a phone is difficult.
     
    Last edited: Aug 16, 2010
  4. Aug 16, 2010 #3

    vela

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    Yes, that works.
     
  5. Aug 16, 2010 #4
    Thanks a lot for the help.
     
  6. Aug 17, 2010 #5
    So what have you decided, does it converges or diverges? :)
     
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