Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By Plato

Math Help - Limit of sequence {b_n}/n --> b

  1. #1
    Newbie
    Joined
    Sep 2012
    From
    usa
    Posts
    2

    Limit of sequence {b_n}/n --> b

    This should be easy, but I missed a week of my real analysis class and the problem recommends using a theorem not in the text.



    Let {bn} be a sequence in IR with the property that bn+1 − bn → b as
    n → ∞. Prove that bn/n → b as n → ∞. [Hint: Use Theorem 3 from
    class].
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1584
    Awards
    1

    Re: Limit of sequence {b_n}/n --> b

    Quote Originally Posted by mickyduu View Post
    This should be easy, but I missed a week of my real analysis class and the problem recommends using a theorem not in the text.
    Let {bn} be a sequence in IR with the property that bn+1 − bn → b as
    n → ∞. Prove that bn/n → b as n → ∞. [Hint: Use Theorem 3 from
    class].
    Shame on you for missing class. It would be nice to know what is th. 3.
    There is a theorem of means:
    If (a_n)\to L then \left( {\frac{1}{n}\sum\limits_{k=1}^n {a_k } } \right) \to L
    Using that we can define a_n=b_n-b_{n-1}-b (if necessary define b_0=0).
    Now clearly (a_n)\to 0 so \left( {\frac{1}{n}\sum\limits_{k=1}^n {a_k } } \right) \to 0.
    But \left( {\frac{1}{n}\sum\limits_{k = 1}^n {a_k } } \right) = \frac{{b_n }}{n} - b.
    Because you missed class and because I have no idea about Th3, that proof may not work.
    Last edited by Plato; October 23rd 2012 at 03:10 PM.
    Thanks from FernandoRevilla
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2012
    From
    usa
    Posts
    2

    Re: Limit of sequence {b_n}/n --> b

    Thanks for the help, but I don't think that your answer works, or more precisely I don't think that it answers the question. I'm pretty sure that I need to show that the


    bn/n → b as n → ∞

    where the bn is a term of the ORIGINAL {bn} sequence.

    My guess is that I'm going to need to start with the definition of a limit for a convergent sequence:

    "A real number x is the limit of the sequence (xn) if the following condition holds:for each ε > 0, there exists a natural number N such that, for every , we have .In other words, for every measure of closeness ε, the sequence's terms are eventually that close to the limit. The sequence (xn) is said to converge to or tend to the limit x, written or."


    I believe that theorem 3 is going to be a common analysis theorem, possibly one that is better known in some generalized form or when dealing with functions2nd +, however our text merely glosses over dealing with sequences.




    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1584
    Awards
    1

    Re: Limit of sequence {b_n}/n --> b

    Quote Originally Posted by mickyduu View Post
    Thanks for the help, but I don't think that your answer works, or more precisely I don't think that it answers the question. I'm pretty sure that I need to show that the
    bn/n → b as n → ∞
    Yes it does. PLease see my EDIT. There was a cut&paste error.

    It proves that \left( {\frac{{b_n }}{n} - b} \right) \to 0 so \left( {\frac{{b_n }}{n}} \right) \to b
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: October 26th 2010, 10:23 AM
  2. Limit of a sequence
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: June 8th 2010, 06:32 AM
  3. Limit of Sequence - max(a,b)
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 25th 2009, 02:18 PM
  4. Limit of a sequence...
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 17th 2009, 06:00 PM
  5. Limit of Sequence
    Posted in the Calculus Forum
    Replies: 7
    Last Post: September 26th 2006, 09:28 AM

Search Tags


/mathhelpforum @mathhelpforum