1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What does the limit imply here?

  1. Jun 6, 2009 #1
    S'ppose this statement:

    [tex] p_1, p_2, p_3,... \in \mathbb {N}[/tex]

    I do understand that p-series is infinite (from the dots) and that every p from the series is a natural number.

    However, does the statement also imply that there is no particular order in the series? I.e. is it possible that

    [tex]p_1 = 3,
    p_2 = 66,
    p_3 = 1[/tex]

    Does the above statement imply that there is no restriction that some p or even all of them are equal? I.e.

    [tex]p_1 = 3, p_2 = 3, p_3 = 12[/tex]


    If all of the above is true, then what does this mean:


    [tex] p_1, p_2, p_3,... \in \mathbb {N}[/tex]

    [tex]
    \lim _{n \to \infty} p_n = \infty[/tex]

    Does the addition of limit statement imply some sort of order in the series?

    Oh, and how do I make new line in latex? \\ and \newline don't seem to work.
     
  2. jcsd
  3. Jun 6, 2009 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    Yes, you understood it correctly. Such a list of numbers p is what we usually call a sequence, and indeed its elements can be anything.

    In special cases, sequences may have a limit. For example, if
    [tex]\lim_{n \to \infty} p_n = L[/tex]
    then we mean that if we make n larger and larger, p(n) will get closer and closer to L. Of course, if p(n) can only take integer values, this means that all the p(n) are equal to L for n sufficiently large. If you substitute L for infinity, we mean that the sequence is unbounded. You should really see this as a convention, although it looks a bit like a limit: we can get the values of the sequence "closer and closer to infinity" as n gets bigger and bigger. More correctly: if we make n bigger and bigger then p(n) will get bigger and bigger -- conversely: we can get p(n) bigger than any number we want by choosing n sufficiently large.

    Note that
    (1): [tex]\lim_{n \to \infty} p_n = \infty[/tex]
    and
    (2): "the limit does not exist"
    are two very different statements. For example, the sequence
    [tex]p_n = n[/tex]
    satisfies (1), while
    [tex]p_n = (-1)^n[/tex]
    satisfies (2).
     
  4. Jun 6, 2009 #3
    Ok, so the limit statement does not imply that p(1)=1, p(2)=2, etc. In other words it does not imply a particular sequence, just a sequence where p(n) gets larger as n gets larger.

    I just want to know what the limit statement in combination with the definition of sequence eliminates. Does it eliminate that for some n p(n)>p(n+1)? Does it eliminate that for some n p(n) = p(n+1)?

    If I understood correctly, the limit statement implies that p(n+1) > p(n) for every n, right? however it does not imply what the jump is from p(n) to p(n+1). it could be that p(1) = 1, p(2) = 13, p(3) = 1000, right? That the sequence is indeed increasing but that the magnitude of each jump is not fixed. Is that right?
     
  5. Jun 6, 2009 #4

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    Indeed.

    Neither. If you give me some integer n, I can always construct a sequence going to infinity and having p(n) > p(n + 1) for that specific n (just define p(k) = k for all k not equal to n, and p(n) = n + 2).

    Not for every n. It does imply that there is some (possibly extremely large) number N, such that p(n + 1) > p(n) whenever n > N (i.e. from a certain point the sequence must be increasing). In fact, that's very nearly the definition:
    [tex]\lim_{n \to \infty} p(n) = \infty[/tex]
    means that for any L there exists N, such that
    whenever n > N, p(n) > L.
     
  6. Jun 6, 2009 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Be careful here. If [itex]\lim_{m\rightarrow \infty} p_n= \infty[/itex] that means [itex]p_n[/itex] eventually becomes larger than any given real number. It does NOT mean that it does that in any simple way! For example, [itex]p_n= n[/itex] for n even, [itex]p_n= n-2[/itex] for n odd gives the sequence -1 2, 1, 4, 3, 6, 5, 8, 7, etc. That "goes to infinity" but I don't think we would say "[itex]p_n[/itex] gets larger as n gets larger" since for every even n, the next number is smaller.

    No, neither of those things.

    No, it does NOT! Saying "[itex]p_{n+1}> p_n[/itex]" for all n is simply saying that [itex]\{p_n\}[/itex] is an "increasing" sequence which may go to infinity or converge to a finite number.

     
  7. Jun 6, 2009 #6

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    As an example of the latter, consider the following sequence of rational number (I don't think an example exists for natural numbers);
    [tex]p_n = 1 - 1/n[/tex]
    The first terms are then 0, 1/2, 3/4, 7/8, ...
    The sequence always increases, but the limit is 1, not infinity.
    Also note that the limit is a "real" limit in the sense that there does not exist an n such that pn = 1. However, by definition of the limit, you can get arbitrarily close: if you tell me how close you want to get to 1 (for example: within 0,001) I can give you an n which realizes that, i.e. pn will be and remain that close to the limit 1 (for 0,001, any n bigger than 1000 will do). In case the "limit" is infinity you need to replace "close to the limit" by something more sensible that expresses that we mean: the sequence is unbounded. In fact, the exact formulation is then: if you tell me how large you want the sequence to get (for example, bigger than 1000000) I can give you an n such that pn is bigger than and remains bigger than 1000000.
     
  8. Jun 6, 2009 #7
    Thanks a lot guys, very clear explanations.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: What does the limit imply here?
  1. Whats wrong here? (Replies: 10)

  2. What is mean D here (Replies: 2)

Loading...