Results 1 to 2 of 2

Math Help - Differentiation Analysis

  1. #1
    Member
    Joined
    Jan 2007
    Posts
    114

    Differentiation Analysis

    Hi,
    Let f: R -> R and differentiable on R. Suppose sup{|f'(x)|} = M < 1. Take S ele R and define a sequence Sn+1 = f(Sn) n = 0,1,2,3,... Show that {Sn} is a cauchy sequence. Hint show first that |Sn+1 - Sn|<= M|Sn - Sn-1|.

    So I know you can start with the mean value theorem: f'(x) = (Sn+1 - Sn)/(Sn - Sn-1). Then (Sn - Sn-1)f'(x) = Sn+1 - Sn.

    From here i'm not exactly sure what to do.
    Thanks

    For those of you who have the kenneth ross book it is exercize 29.18.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Theorem: If 0<a<1 and |s_{n+1}-s_n| \leq a|s_n - s_{n-1}|. Then the sequence is convergent (or Cauchy).

    Proof: Here.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Analysis: Absolute Continuity & Differentiation
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 22nd 2010, 05:50 AM
  2. urgent- Analysis differentiation
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 31st 2009, 08:46 AM
  3. VERY hard differentiation analysis questions
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: May 31st 2009, 08:08 AM
  4. Analysis differentiation
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 30th 2009, 04:36 PM
  5. Differentiation (Real Analysis)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 30th 2008, 06:32 PM

Search Tags


/mathhelpforum @mathhelpforum