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The big O notationm, what does it mean?

  1. Nov 10, 2008 #1
    1. The problem statement, all variables and given/known data

    Do you know what O(x) means? This O was put after a Taylor polynomial, and I don't get what it means. I've read the definition, and I don't really get it, what does it actually tell me?

    For example, what does writing exp(x) = 1 + x + x^2/4 + O(x^3) , where Rn(x) is O(x^3) (Rn(x) is the remainder) tell me?
    Last edited: Nov 11, 2008
  2. jcsd
  3. Nov 10, 2008 #2


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    I feel a compelling need to point out it's x2/2 actually. Regardless.

    If you have f(x) = o(p(x)) for some f and p, it means that as x goes to zero, f goes to zero faster than p, i.e. f/p -> 0 as x goes to zero. This is meant to indicate that near zero something is so small, it's going to vanish when you take a limit to zero.

    O is the opposite. f(x) = O(p(x)) means that as x goes to infinity, f is bounded above by p, i.e. there is some constant C such that f(x) <= C*p(x) for all large x.

    So in your Taylor series example, I don't agree with what's written (assuming you meant little o as you stated in your second sentence) as the remainder is

    [tex]x^3/6 + ...[/tex] and when you divide through by x3 you get [tex]1/6 + x*(stuff that's at least constant)[/tex] which doesn't go to zero. It IS o(x2 as when you divide through by x2 you get

    [tex]x/6 + x^2/24 + ... = x*(stuff)[/tex] and that goes to zero as x goes to zero
  4. Nov 11, 2008 #3
    Re: The big O notation, what does it mean?

    Yes, it's 2! = 2 :D

    Anyway, it's a big O, not a little o, I don't know why I wrote little o, I was tired I guess. Sorry. I mean big O.
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