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Question on Big-O notation

  1. Mar 13, 2012 #1

    I have a polynomial having the form:

    [tex]\beta x^5 + \beta^2 x^7 + \beta^3 x^9 + \ldots = \sum_{n=1}^{+\infty}\beta^n x^{2n+3}[/tex]

    How can I express this with Big-O notation?
    Please, note that I consider β as another variable (independent from x). I already know that if β was a constant I could express the above quantity as [itex]O(x^5)[/itex].
  2. jcsd
  3. Mar 13, 2012 #2


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  4. Mar 13, 2012 #3
    First of all I forgot to mention that [itex]x\geq 0[/itex] and [itex]\beta \in \mathbb{R}[/itex].

    I will try to apply the definition found in Wikipedia, although that definition refers specifically to functions of one-variable. I am not sure we can use that definition, but I will try.

    Let's "pretend" that β is a constant and x the variable. We have: [tex]f(x)=\beta x^5 + \beta^2 x^7 + \beta^3 x^9\ldots[/tex]

    and I am interested in studying the behavior for [itex]x\to 0[/itex].
    We have that: [tex]|\beta x^5 + \beta^2 x^7 + \beta^3 x^9\ldots | \leq |\beta| x^5 + |\beta^2| x^7 + |\beta^3| x^9\ldots \leq |\beta| x^5 + |\beta^2| x^5 + |\beta^3| x^5 \ldots [/tex], hence we have [itex]f(x)\in O(r^5)[/itex], as expected.

    By considering β variable, and x constant we have [itex]f(\beta) \in O(\beta)[/itex].

    Now what?
  5. Mar 13, 2012 #4
    So you want x->0 and [itex]\beta \to 0[/itex] right?

    Your series looks a lot like a geometric series. In fact
    [tex]\sum_{i=1}^\infty \beta^nx^{3+2n} = x^3\sum_{i=1}^\infty (\beta x^2)^n[/tex]
    For small enough [itex]\beta[/itex] and x you should get a nice closed form from which you can more easily see the series' asymptotic behavior.
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