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Convergence of a series

  • #1

Homework Statement


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 ?


Homework Equations



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

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 ?
 

Answers and Replies

  • #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:
  • #3
vela
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Yes, that works.
 
  • #4
Thanks a lot for the help.
 
  • #5
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So what have you decided, does it converges or diverges? :)
 

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