Results 1 to 3 of 3

Math Help - Prove the limit of this sequence exists

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    60

    Prove the limit of this sequence exists

    Hi, I'm having a little bit trouble applying the Monotone Sequence Theorem to this question:

    Determine if the sequence defined by the recursive relation:

    x_{k+1} = \sqrt{2+3x_k}
    x_1=1

    has a limit. Present your reasoning by using the Monotone Sequence Theorem.

    Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Use induction to show that x_{n}\le x_{n+1}\le 4.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The difference equation that defines the sequence can be written as...

    \displaystyle \Delta_{n} = x_{n+1}-x_{n} = \sqrt{2+3\ x_{n}}-x_{n} = f(x_{n}) (1)

    The function f(*) is represented here...



    There is only one 'attractive fixed point' in x_{0}=3.5615528128 \dots and, because for any x\ge -\frac{3}{2} is |f(x)| < |x_{0}-x| , any x_{1}\ge -\frac{3}{2} will produce a sequence that converges monotonically at x=x_{0}...

    Kind regards

    \chi \sigma



    Kind regards

    \chi \sigma
    Last edited by chisigma; October 14th 2010 at 11:12 PM. Reason: The post has been modified...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to prove such a sequence exists
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 6th 2011, 08:49 AM
  2. How to prove that this limit does exists...
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 26th 2011, 07:51 AM
  3. How to prove that a limit exists at 0.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 27th 2011, 02:22 PM
  4. [SOLVED] How to prove that a limit exists?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 1st 2010, 06:02 AM
  5. limit, sequence, exists
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 15th 2009, 11:01 PM

Search Tags


/mathhelpforum @mathhelpforum