Results 1 to 3 of 3

Math Help - Find the superior and inferior limit??

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    6

    Find the superior and inferior limit??

    Hi guys:
    the Qs is 'Find the superior and inferior limit of the sequence"(-1)^n(1+n^-1)". Even I know the answer for the Qs is 1 and -1, but what should be the currect process for this Qs, please give some detail.thanks!! <img smilieid="84" class="inlineimg" src="http://www.mathhelpforum.com/math-help/images/smilies/Skype/emoticon-0139-bow.gif" border="0">
    Follow Math Help Forum on Facebook and Google+

  2. #2
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    Quote Originally Posted by wsun View Post
    Hi guys:
    the Qs is 'Find the superior and inferior limit of the sequence"(-1)^n(1+n^-1)". Even I know the answer for the Qs is 1 and -1, but what should be the currect process for this Qs, please give some detail.thanks!! <img smilieid="84" class="inlineimg" src="http://www.mathhelpforum.com/math-help/images/smilies/Skype/emoticon-0139-bow.gif" border="0">
    I give you a theorem to judge the superior limit of a sequence.
    Let E=\{a|\exists subsequence\{b_n\}, \lim b_n=a\},and +\infty,-\inftymay also appear in E.
    Then bis the superior limit of \{b_n\}iff
    1: b\in E
    2: \forall x>b, \exists N, \forall n>N, a_n<x
    For the inferior limit, it is similiar.
    Now you can easily identify whether the answer is 1 or -1..
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,696
    Thanks
    1467
    Your sequence is given by a_n= (-1)^n(1+ \frac{1}{n}). Calculate a few values: a_1= -2, a_2= \frac{3}{2}, a_3= -\frac{4}{3}, a_4= \frac{5}{4}.

    In fact it should be clear that a_n= (-1)^n(1+ \frac{1}{n})= (-1)^n\frac{n+1}{n}. If n is odd that is a_n= -\frac{n+1}{n} which converges to -1 and if n is even that is a_n= \frac{n+1}{n} which converges to 1. Since, if a sequence converges, every convergent subsequence must converge to the same thing, it is clear that every convergent subsequence converges to either 1 or -1.

    Now, "lim sup" is defined as the supremum (least upper bound) of the set of subsequential limits while the "lim inf" is the infimum. Here, the set of subsequential limits is {-1, 1} which has minimum -1 and maximum, 1 and, since it has a minimum and maximum, those are the "inf" and "sup" and so the lim inf and lim sup.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Limit Superior and Limit Inferior
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 11th 2011, 04:43 AM
  2. limit superior/inferior proof
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: February 10th 2011, 04:36 AM
  3. limit superior and limit inferior
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 17th 2010, 07:28 PM
  4. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 3rd 2009, 05:05 PM
  5. Replies: 2
    Last Post: October 4th 2008, 04:57 PM

Search Tags


/mathhelpforum @mathhelpforum