Results 1 to 3 of 3

Math Help - Convergence

  1. #1
    Newbie
    Joined
    Jul 2006
    Posts
    12

    Convergence

    Let yn = the sqrt(n+1)-sqrt(n).
    Prove that both yn and sqrt(n)yn converge.

    Could someone help me?
    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 OntarioStud View Post
    Let yn = the sqrt(n+1)-sqrt(n).
    Prove that both yn and sqrt(n)yn converge.

    Could someone help me?
    We note that,
    {y_n} is a postive sequence because,
    n+1>n thus, sqrt(n+1)>sqrt(n) thus, sqrt(n+1)-sqrt(n)>0
    Thus it has a lower bound.

    The sequence is also strictly decreasing.
    Because,
    y_{n+1}<y_n
    If and only if,
    y_{n+1}-y_n<0
    If and only if,
    sqrt(n+2)-sqrt(n+1)-sqrt(n+1)+sqrt(n)<0
    If and only if,
    sqrt(n+2)-sqrt(n)<2sqrt(n+1)
    Which is true (square both sides).

    Thus, by Weierstrauss-Bolzano Theorem this sequence has a limit.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    For the second problem:

    We have,
    y_n=sqrt(n+1)-sqrt(n)
    Thus,
    x_n=sqrt(n)y_n=sqrt(n^2+n)-n
    We can show that,
    x_{n+1}>x_n
    Thus it is strictly increasing.

    Then we show using induction that,
    x_n<1

    Again, it must converge.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: May 13th 2010, 01:20 PM
  2. Replies: 2
    Last Post: May 1st 2010, 09:22 PM
  3. dominated convergence theorem for convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 5th 2009, 04:06 AM
  4. Replies: 6
    Last Post: October 1st 2009, 09:10 AM
  5. Pointwise Convergence vs. Uniform Convergence
    Posted in the Calculus Forum
    Replies: 8
    Last Post: October 31st 2007, 05:47 PM

Search Tags


/mathhelpforum @mathhelpforum