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What is a limit of a sequence?

  1. Feb 16, 2006 #1
    for lim n to infinity {2 [(-1)^n] },what should i write either converges or diverges because it tends to be -2 or +2
     
  2. jcsd
  3. Feb 16, 2006 #2

    arildno

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    What is a limit of a sequence?
     
  4. Feb 16, 2006 #3
    limit n to infinity
     
  5. Feb 16, 2006 #4
    If i'm not mistaken, first thing you need to check in a series if the limit is unique , as in when the series limit is one number.
    then you should check if that limit is zero or not , then you can apply usual methodes know to find out.
    So in your case it diverge.
     
  6. Feb 16, 2006 #5
    is it if there is (-1^n) and n limit to infinity means always diverges??
     
  7. Feb 16, 2006 #6
    doesn't have to.
    [(-1)^n]/n have a limit when n tends to be infinite 0 , this series converge.
    I sense you don't know much about limits , maybe you didn't studied well that chapter.
    it's fairly easy to know them.
     
  8. Feb 16, 2006 #7

    arildno

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    Again, teng:
    What is the definition of a limit?
     
  9. Feb 17, 2006 #8
    "limit" is how a function's behavior changes when its argument(or variable) gets close to a certain value(or a point)
     
  10. Feb 17, 2006 #9

    arildno

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    Not at all.
     
  11. Feb 17, 2006 #10
    why is that?
     
  12. Feb 17, 2006 #11

    arildno

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    Because what you wrote is meaningless.
    Go back to your textbook and look up the definition of a limit to a sequence.
     
  13. Feb 17, 2006 #12
    but ur question simply says
    arildno :
    What is the definition of a limit?

    u dint specify it was limit to a sequence...
     
  14. Feb 17, 2006 #13
    i just thought u were saying abt limit of a function....so sorry
     
  15. Feb 17, 2006 #14

    VietDao29

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    Nah, this is neither the definition for limit of a function nor limit of a sequence...
    You may want to look it up in your book. :)
    By the way, are you teng125?
     
  16. Feb 17, 2006 #15

    arildno

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    Besides, a sequence IS a function.
     
  17. Feb 17, 2006 #16

    benorin

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    OK, the lim inf is -2 and the lim sup is +2, and thus the lim is...
     
  18. Feb 18, 2006 #17

    arildno

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    teng:
    Forget subsequences, lim infs and all that.
    Those concepts won't help you a bit, because you betray an uncertainness abut the very concept of a limit in the first place.

    Let us take a typical textbook definition:
    "We say that a number L is a LIMIT of a sequence [itex]a_{n}[/tex] if for any [itex]\epsilon>0[/itex] there exists a number N, so that for any n>N, [itex]|a_{n}-L|<\epsilon[/itex]"
    Furthermore, a sequence is said to diverge if no such number L exists.

    1. The first thing to note is that L (if it exists) is a NUMBER, and only that.
    It is not a hand-wavy action by which we describe the function's behaviour. It is a number. Period.

    2. Secondly, the definition should be regarded as a recipe of detemining whether or not an arbitrarily chosen L is a limit to the sequence or not.

    3. Let us for convenience sake pick L=2 first, and check whether it can be said to be a limit to the sequence:

    Consider the absolute valued difference: [tex]|2(-1)^{n}-2|[/tex]
    Now, when n is even, this difference equals 0, but when n is odd, the difference equals 4.
    Thus, by picking [itex]\epsilon<4[/itex], and remembering that odd numbers can be arbitrarily big, we see that there cannot exist a number N so that for ANY n>N, the difference is less than [itex]\epsilon[/itex]

    Thus, 2 cannot be regarded as a limit L to our sequence.


    Do you think -2 can be a limit?
    What about any other number?
    Answer those questions, and you have the answer to your own question.
     
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