Results 1 to 3 of 3

Math Help - Prove limit of recurrence relation

  1. #1
    Newbie
    Joined
    Feb 2007
    Posts
    23

    Prove limit of recurrence relation

    Let S1 = sqrt(6), S2 = sqrt( 6 + sqrt(6) ), S3 = sqrt( 6 + sqrt(6) +sqrt(6) ), and in general define S(n+1) = sqrt( 6 + Sn ). Prove that Sn converges, and find its limit
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by slowcurv99 View Post
    Let S1 = sqrt(6), S2 = sqrt( 6 + sqrt(6) ), S3 = sqrt( 6 + sqrt(6) +sqrt(6) ), and in general define S(n+1) = sqrt( 6 + Sn ). Prove that Sn converges, and find its limit
    Hint: Show it is a bounded monotone sequence.

    Hint2: lim s_{n+1} = lim s_n
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,825
    Thanks
    714
    Hello, slowcurv99!

    Let S1 = sqrt(6), S2 = sqrt[6 + sqrt(6], S3 = sqrt{6 + sqrt[6 + sqrt(6)]},

    and in general define: .S
    n+1 .= .sqrt(6 + Sn).

    Prove that S
    n converges, and find its limit.

    We have: .S .= .sqrt
    {6 + sqrt[6 + sqrt(6) + ... ]}
    . . . . . . . . . . . . . . . . . . .\_________________/
    . . . . . . . . . . . . . . . . . . . . . . .
    This is S
    . . . . . . . . . . . . . . . . . . ____
    Hence, we have: . S .= .√6 + S

    Square both sides: . .= .6 + S . . Sē - S - 6 .= .0

    . . which factors: .(S - 3)(S + 2) .= .0

    . . and has roots: .S = 3, -2


    Taking the positive root: .S = 3

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove that recurrence relation is odd
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: September 14th 2011, 02:24 AM
  2. Prove recurrence relation equals sum of permutations
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: July 16th 2010, 09:00 AM
  3. How to find the limit of recurrence relation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 30th 2009, 07:36 AM
  4. Recurrence Relation
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: January 13th 2009, 03:55 PM
  5. Recurrence Relation
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: January 13th 2009, 03:37 PM

Search Tags


/mathhelpforum @mathhelpforum