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Math Help - liminf, limsup, limit

  1. #1
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    liminf, limsup, limit

    Hi... I am trying to understand liminf and limsup of sequences.

    If I have the sequence

    {\frac{1+k}{1+e^k}} {+ sin(k)}<br />

    what is my liminf and my limsup?

    The limit itself does not exist, so the two should necessarily be unequal, right?

    I know that

    limsup=inf_{k \ge 1} sup_{n \ge k} {\frac{1+n}{1+e^n}} +sin(n)

    but how do I determine the sup, if it exists?

    Also, for the sequence

    {(2+(-1)^k)^k},

    am I correct that neither the limit nor the limsup exist but that {liminf=1}?

    And finally, for the sequence

    {1 + {\frac{(-1)^k}{k^2}}}

    because the {lim=1}, that means that {liminf=limsup=1}, right?

    Many thanks!
    Last edited by TXGirl; May 14th 2008 at 05:32 PM. Reason: LaTeX
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  2. #2
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    Hi TXGirl
    You need to learn to use LaTex so that we can read your questions.
    I know that is is a real pain, but it is well worth it.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Hi TXGirl
    You need to learn to use LaTex so that we can read your questions.
    I know that is is a real pain, but it is well worth it.

    It's true.... I read a tutorial and changed the posting....

    LaTeX Tutorial
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  4. #4
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    Quote Originally Posted by TXGirl View Post
    Hi... I am trying to understand liminf and limsup of sequences.

    If I have the sequence

    {\frac{1+k}{1+e^k}} {+ sin(k)}<br />

    what is my liminf and my limsup?

    The limit itself does not exist, so the two should necessarily be unequal, right?

    I know that

    limsup=inf_{k \ge 1} sup_{n \ge k} {\frac{1+n}{1+e^n}} +sin(n)

    but how do I determine the sup, if it exists?

    Also, for the sequence

    {(2+(-1)^k)^k},

    am I correct that neither the limit nor the limsup exist but that {liminf=1}?

    And finally, for the sequence

    {1 + {\frac{(-1)^k}{k^2}}}

    because the {lim=1}, that means that {liminf=limsup=1}, right?

    Many thanks!
    Anybody have any ideas?
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