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Homework Help: Liminf and limsup problem

  1. Mar 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Find liminf(xn) and limsup(xn) for xn = n(1-(-1)^n)


    2. Relevant equations



    3. The attempt at a solution
    I'm not really getting liminf and limsup, but stumbled through the method in my textbook and got liminf=0, limsup doesn't exist.

    Is that right? I don't think I did it right. If it's wrong, I can type out my working (I'd just prefer not to since it's pretty late).
     
  2. jcsd
  3. Mar 4, 2010 #2
    You are correct. I fins the easiest definition of limsup to be:

    [tex]\limsup_{n\rightarrow \infty}=\lim_{n\rightarrow \infty}(\sup_{m\geq n}x_m)[/tex]

    That is: as n approaches infinity, look at all the "peaks" of the [tex]x_m[/tex] for [tex]m\geq n[/tex]. You are looking for the "last peak".

    In the case of this function, the peaks get larger and larger (they occur when n is odd), so the "last one" is infinity, ie. it doesn't exist.

    Similarly for liminf, the "troughs" of the function are all zero (when n is even), so the "last one" will also be zero.

    I don't know if this will help much...I remember also being very confused by liminf and limsup when I first came across them!
     
  4. Mar 4, 2010 #3
    It helped a lot, thanks!
     
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