Results 1 to 2 of 2

Math Help - Asymptotic equivalent of a sequence

  1. #1
    Newbie Hugal's Avatar
    Joined
    Oct 2010
    From
    The Farthest Land
    Posts
    16

    Asymptotic equivalent of a sequence

    Hi everyone,

    I've a little trouble with a small exercice.
    So, given the equation x^n+x=1, it can ben easily shown that \displaystyle \exists ! x_n \in \mathbb{R}^+ / x_n^n + x_n = 1.

    I also prooved that the sequence (x_n)_{n > 0} is convergent and its limit is l=1.

    Now, the question is : Find an asymptotic equivalent of x_n - l. And this is what I have trouble with.

    I've tried some thing, for example, we know that x_n - 1 = - x_n^n but this expression do not seem to bring something good.
    So, I don't want the whole answer, but just the little trick that I've not seen and which can end this question.

    Thanks for reading me, and sorry if my english is little bit bad,

    Hugo.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: Asymptotic equivalent of a sequence

    Hi,
    we have x_n = 1-x_n^n = (1-x_n)\sum_{k=0}^{n-1}x_n^k hence \displaystyle 1-x_n = \dfrac{x_n}{\sum_{k=0}^{n-1}x_n^k} = \dfrac{x_n(1-x_n)}{1-x_n^n}. Since \lim_{n\to\infty}x_n=1 we have 1-x_n\sim\frac{1-x_n}{1-x_n^n}. Now, use the mean value theorem to show that an asymptotic equivalent of 1-x_n is \frac 1n.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Asymptotic Variance
    Posted in the Statistics Forum
    Replies: 1
    Last Post: January 5th 2012, 07:35 AM
  2. Asymptotic
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 30th 2011, 12:02 AM
  3. asymptotic bounds
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: May 8th 2010, 05:20 PM
  4. asymptotic relations
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 4th 2008, 04:20 AM
  5. help for asymptotic expansion
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 3rd 2007, 03:31 AM

Search Tags


/mathhelpforum @mathhelpforum