Results 1 to 2 of 2

Math Help - cauchy sequence

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    172

    cauchy sequence

    Prove that the sequence :
    x_n = n^2 + \frac{(-1)^n}{n} is not cauchy
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by flower3 View Post
    Prove that the sequence :
    x_n = n^2 + \frac{(-1)^n}{n} is not cauchy


    If it were Cauchy then for any \epsilon >0\,\,\,\exists M\in\mathbb{N}\,\,\,s.t.\,\,\,|x_n-x_m|<\epsilon\,\,\,\forall\,n,m>M . In particular, this must be true for m=n+1 , but:

    |x_n-x_{n+1}|=\left|-2n-1+\frac{(-1)^n}{n(n+1)}\right|=2n+1\pm \frac{1}{n(n+1)}>2 , so it is enough to take \epsilon < 2 and the above definitory property of Cauchy sequences won't be true for it.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. cauchy sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 1st 2010, 05:40 AM
  2. [SOLVED] Subsequence of a Cauchy Sequence is Cauchy
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: September 30th 2010, 01:29 AM
  3. cauchy sequence, contractive sequence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 25th 2010, 06:25 AM
  4. cauchy sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 3rd 2009, 06:06 PM
  5. Cauchy sequence
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 24th 2006, 01:03 PM

Search Tags


/mathhelpforum @mathhelpforum