Results 1 to 4 of 4

Math Help - limsup and liminf

  1. #1
    Junior Member
    Joined
    Sep 2006
    Posts
    32

    limsup and liminf

    I have to find the limit superior and the limit inferior of these sequences:

    a)
    n + [(-1)^n](2n+1)
    ---------------- It appears limsup = 3 and liminf = -1.
    n

    b)
    (an)^(1/n) where an = 1/(2^[(n+1)/2]) if n is odd and 1/(3^(n/2)) if n is even. What are the limsup and liminf of these? They seem less obvious.

    Thanks in advance for any help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by MKLyon View Post
    I have to find the limit superior and the limit inferior of these sequences:

    a)
    n + [(-1)^n](2n+1)
    ---------------- It appears limsup = 3 and liminf = -1.
    n
    i agree


    b)
    (an)^(1/n) where an = 1/(2^[(n+1)/2]) if n is odd and 1/(3^(n/2)) if n is even. What are the limsup and liminf of these? They seem less obvious.

    Thanks in advance for any help.
    we have: a_n = \left \{ \begin{array}{lr} \frac 1{2^{(n + 1)/2}}  , & \mbox{ for } n \mbox { odd} \\ & \\  \frac 1{3^{n/2}} , & \mbox{ for } n \mbox{ even} \end{array}\right.

    thus (a_n)^{\frac 1n} = \left \{ \begin{array}{lr} \frac 1{2^{(n + 1)/2n}}  , & \mbox{ for } n \mbox { odd} \\ & \\ \frac 1{\sqrt{3}} , & \mbox{ for } n \mbox{ even} \end{array}\right.

    can you take it from here?

    consider when n = 1 as well as when n \to \infty
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2006
    Posts
    32
    So 1/sqrt(2) and 1/sqrt(3)?

    Thanks for responding.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by MKLyon View Post
    So 1/sqrt(2) and 1/sqrt(3)?

    Thanks for responding.
    did you consider when n = 1? what happens then?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove limsup-liminf<2
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 6th 2011, 04:15 PM
  2. Liminf/Limsup of a Sum of Sequences
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 4th 2010, 11:00 PM
  3. [SOLVED] limsup and liminf
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 21st 2010, 06:09 AM
  4. Liminf to limsup
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 20th 2009, 11:49 AM
  5. A problem with limsup/liminf
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 8th 2009, 02:20 PM

Search Tags


/mathhelpforum @mathhelpforum