Results 1 to 4 of 4

Math Help - Recursive Sequence Convergence

  1. #1
    Newbie
    Joined
    Oct 2010
    Posts
    5

    Recursive Sequence Convergence

    x_0 > x_1 and r and s are positive numbers with p + q = 1. x_n = px_{n-1} + qx_{n-2} for all n \geq 2. Prove that (x_n) is convergent and find limits in terms of x_0, x_1, p and q.

    Okay I began by letting p = 1 - q so I could eliminate that variable. Then I began working it out up to x_7 but I couldn't find any pattern in terms of coefficients of the polynomial. It's a polynomial with respect to p. I am trying to get this to work out to something that I can put into summation notation, prove it is monotone, and then find an upper and/or lower bound but I am unsure how to go about putting it in summation notation - it's entirely possible this is not the way to do it but it seems most logical to me. The first term (in order of descending exponent on p) of the expansion is always p^{n-1}*a_1. The last term is a_0 when n is even and a_1 when n is odd. These are the only real patterns I've been able to discern though. Any insight is much appreciated. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Here is an idea for you The sequence can be written as follows

    \begin{bmatrix}0 && 1 \\ (1-p) && p \end{bmatrix}\begin{bmatrix} x_{n-2} \\x_{n-1}\end{bmatrix}=\begin{bmatrix} x_{n-1} \\x_{n}\end{bmatrix}

    This matrix can be diagonalized and solved explicitly

    For the two eigenvalues I got

    \lambda = p-1 \text{ or } \lambda =1

    I hope this helps.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by valerian View Post
    x_0 > x_1 and r and s are positive numbers with p + q = 1. x_n = px_{n-1} + qx_{n-2} for all n \geq 2. Prove that (x_n) is convergent and find limits in terms of x_0, x_1, p and q.

    Okay I began by letting p = 1 - q so I could eliminate that variable. Then I began working it out up to x_7 but I couldn't find any pattern in terms of coefficients of the polynomial. It's a polynomial with respect to p. I am trying to get this to work out to something that I can put into summation notation, prove it is monotone, and then find an upper and/or lower bound but I am unsure how to go about putting it in summation notation - it's entirely possible this is not the way to do it but it seems most logical to me. The first term (in order of descending exponent on p) of the expansion is always p^{n-1}*a_1. The last term is a_0 when n is even and a_1 when n is odd. These are the only real patterns I've been able to discern though. Any insight is much appreciated. Thanks.
    The sequence is defined by the 'recursive relation'...

    \displaystyle x_{n} - p\ x_{n-1} - q\ x_{n-2}=0 (1)

    ... that is 'linear and homogeneous' and the solution of which is...

    \displaystyle x_{n} = c_{1}\ \alpha_{1}^{n} + c_{2}\ \alpha_{2}^{n} (2)

    ... where c_{1} and c_{2} are 'arbitrary constants' that depend from x_{0} and x_{1} , \alpha_{1} and \alpha_{2} the solution of the 'characteristic equation'...

    \displaystyle \alpha^{2}- p\ \alpha - q= 0 \implies \alpha= \frac{p \pm \sqrt{p^{2}+4\ p -4}}{2} (3)

    Now for 0< p= 1-q <1 is |\alpha_{1}|<1 and |\alpha_{2}|<1 , so that in any case is...

    \displaystyle \lim_{n \rightarrow \infty} x_{n} = 0 (4)

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    Put x_n=m^n... and try to solve the quadratic equation... (hmmm...)

    [EDIT: 4 MINUTES LATE]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Determine the convergence of a recursive sequence
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 26th 2010, 10:41 PM
  2. Proof of Convergence for Recursive Sequence
    Posted in the Calculus Forum
    Replies: 9
    Last Post: June 4th 2010, 02:59 AM
  3. Replies: 2
    Last Post: March 1st 2010, 12:57 PM
  4. recursive sequence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 29th 2009, 06:15 AM
  5. recursive sequence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 6th 2008, 12:21 AM

Search Tags


/mathhelpforum @mathhelpforum