Results 1 to 2 of 2

Math Help - Infinitesimal 2. Second question!

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Infinitesimal 2. Second question!

    Given x_0 = 3, we define recursively (x_(n)), an infinite sequence starting at n=1 and the recursive formula is:
    x_(n+1) = x_(n) - tan x_(n)

    Prove that the sequence is convergent and find her limit.

    * You may use the fact that: |tan3|<0.1426



    Thank you all!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Also sprach Zarathustra View Post
    Given x_0 = 3, we define recursively (x_(n)), an infinite sequence starting at n=1 and the recursive formula is:
    x_(n+1) = x_(n) - tan x_(n)

    Prove that the sequence is convergent and find her limit.

    * You may use the fact that: |tan3|<0.1426



    Thank you all!
    If this converges it converges to a zero of \tan(x). There is such a zero at x=\pi which is close to your starting value, and a little experimentation indicated that the sequence converges to this zero of \tan(x). Now you just have to prove that it converges.

    You will need to use a result something like:

    |\varepsilon|\le |\tan(\pi+\varepsilon)|<|2\varepsilon|

    for |\varepsilon|<0.5

    CB
    Last edited by CaptainBlack; March 20th 2010 at 12:20 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: May 24th 2011, 05:51 PM
  2. infinitesimal sketch to "show" d/dx (e^ix)
    Posted in the Calculus Forum
    Replies: 7
    Last Post: January 10th 2011, 08:09 PM
  3. Infinite set mapping to infinitesimal interval
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: August 1st 2010, 12:17 AM
  4. Infinitesimal 2. First question! (Hard one...)
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 19th 2010, 07:08 AM

Search Tags


/mathhelpforum @mathhelpforum