Results 1 to 3 of 3

Thread: Show convergence

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    17

    Show convergence

    if a$\displaystyle _{n+2}$ = (a$\displaystyle _{n+1}$ + a$\displaystyle _n$)/2 for all n greater or equal to 1. show that
    a$\displaystyle _n$ converges to (a$\displaystyle _n$ + 2a$\displaystyle _2$)/3.

    Please help me. I know that we want to show that the absolute value of a$\displaystyle _n$ minus(a$\displaystyle _n$ + 2a$\displaystyle _2$)/3 is less than epsilon but I am not sure how to get there. Please help.

    Ok so am i supposed to know what the first few terms are? can somebody please help me?
    Last edited by janae77; Apr 21st 2010 at 06:00 AM. Reason: Re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,802
    Thanks
    3035
    The question itself makes no sense. If $\displaystyle a_n$ is sequence indexed by "n" then its limit cannot involve "n". Are you sure you have copied this correctly?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by janae77 View Post
    if $\displaystyle a_{n+2}=\frac{ a_{n+1} + a_n}{2}$ for all n greater or equal to 1. show that
    $\displaystyle a_n$ converges to $\displaystyle \frac{a_1+2a_2}{3}$.
    (I corrected the question)

    You can notice that the sequence $\displaystyle u_n=a_{n+1}-a_{n}$ satisfies $\displaystyle u_{n+1}=-\frac{1}{2}u_n$ and $\displaystyle u_1=a_2-a_1$ hence by induction $\displaystyle u_{n+1} = \frac{(-1)^n}{2^n}(a_2-a_1)$ (this is a geometric sequence), and you can then compute $\displaystyle a_{n+1}-a_1=u_n+u_{n-1}+\cdots+u_1=\cdots$ (sum of terms from a geometric sequence) and conclude by taking the limit as $\displaystyle n\to\infty$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: Oct 3rd 2011, 01:12 AM
  2. Replies: 4
    Last Post: Apr 19th 2011, 06:39 PM
  3. Replies: 1
    Last Post: Oct 31st 2010, 07:09 PM
  4. Show sequence convergence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 25th 2009, 08:27 PM
  5. sequence, show convergence.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 30th 2009, 01:35 AM

Search Tags


/mathhelpforum @mathhelpforum