Results 1 to 3 of 3

Thread: Monotone Sequences

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    4

    Question Monotone Sequences

    I could really use some help in one of my questions


    Let alpha be a fixed real number such that alpha is strictly less than 1/3. The sequence is defined recusively as follows:

    a1 = alpha an+1=1/4(1+an) for n greater or equal to 1


    (a) Show that an is less than or equal to 1/3for all n. (Use induction on n.)
    (b) Show that (an) is increasing
    (c) Deduce that (an) has a limit and determine lim(an) as n->infinity
    (d) Show that there exists an nsub0 greater or equal to 1 such that an>0 for all n greater or equal to nsub0

    I have done up to (c) but i can't find any notes on part (d). Can anyone help?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,781
    Thanks
    3030
    Well, that is basically, the definition of limit isn't it? If $\displaystyle lim a_n= L$, then given any $\displaystyle \epsilon> 0$, there exist N such that if n> N, $\displaystyle |a_n- L|< \epsilon$. Presumably you got a positive number for the limt in (c) (otherewise (d) is not true!). Use the definition of limit with $\displaystyle \epsilon$ equal to that limit.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2008
    From
    Indiana
    Posts
    127
    Quote Originally Posted by HallsofIvy View Post
    Well, that is basically, the definition of limit isn't it? If $\displaystyle lim a_n= L$, then given any $\displaystyle \epsilon> 0$, there exist N such that if n> N, $\displaystyle |a_n- L|< \epsilon$. Presumably you got a positive number for the limt in (c) (otherewise (d) is not true!). Use the definition of limit with $\displaystyle \color{red}{\epsilon}$ equal to that limit.
    I think you meant to say $\displaystyle L$?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question on monotone sequences
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Aug 2nd 2011, 03:52 AM
  2. Question about monotone sequences.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Jun 30th 2011, 09:44 AM
  3. Monotone Sequences
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 25th 2010, 04:08 PM
  4. Monotone sequences and Cauchy sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Mar 21st 2009, 08:59 PM
  5. monotone sequences
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 10th 2007, 05:36 PM

Search Tags


/mathhelpforum @mathhelpforum