Results 1 to 6 of 6

Math Help - Sequences convergence problem

  1. #1
    Junior Member Kanwar245's Avatar
    Joined
    Jun 2011
    From
    Canada
    Posts
    68
    Thanks
    2

    Sequence convergence problem

    Let a1 > 1, and suppose an+1 = 2 - 1/an for all n >= 2 . Prove that (an) converges, and find its limit.
    Last edited by Kanwar245; March 15th 2013 at 08:43 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,959
    Thanks
    1783
    Awards
    1

    Re: Sequences convergence problem

    Quote Originally Posted by Kanwar245 View Post
    Let a1 > 1, and suppose an+1 = 2 - 1/an for all n >= 2 . Prove that (an) converges, and find its limit.
    It is sufficient to show that the sequence is decreasing and bounded below.
    Do that with induction.

    The solve the equation L=2-\frac{1}{L}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member Kanwar245's Avatar
    Joined
    Jun 2011
    From
    Canada
    Posts
    68
    Thanks
    2

    Re: Sequences convergence problem

    Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,959
    Thanks
    1783
    Awards
    1

    Re: Sequences convergence problem

    Quote Originally Posted by Kanwar245 View Post
    Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2
    Why should anyone think that?

    Take a look at this
    Last edited by Plato; March 15th 2013 at 04:46 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member Kanwar245's Avatar
    Joined
    Jun 2011
    From
    Canada
    Posts
    68
    Thanks
    2

    Re: Sequences convergence problem

    Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that an+1 >= an right
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,959
    Thanks
    1783
    Awards
    1

    Re: Sequences convergence problem

    Quote Originally Posted by Kanwar245 View Post
    Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that an+1 >= an right
    BASE CASE: Prove that a_1>a_2\ge 1.

    PROVE
    Inductive case: If a_k>a_{k-1}\ge 1 then a_{k+1}>a_k\ge 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sequences convergence or not
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 24th 2012, 07:56 PM
  2. Sequences, convergence
    Posted in the Calculus Forum
    Replies: 6
    Last Post: September 2nd 2011, 03:37 AM
  3. Convergence of Sequences
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: March 6th 2011, 07:06 AM
  4. Convergence in sequences of sequences
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 19th 2010, 08:28 AM
  5. sequences/convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 3rd 2007, 12:31 PM

Search Tags


/mathhelpforum @mathhelpforum