Question on Big-O notation

  • Thread starter mnb96
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  • #1
713
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Hello,

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].
 

Answers and Replies

  • #3
713
5
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?
 
  • #4
430
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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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