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Question on BigO notation 
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#1
Mar1312, 05:16 AM

P: 625

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 BigO 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
Mar1312, 05:34 AM

P: 4,573

Hey mnb96.
My answer was going to be based on this page: http://en.wikipedia.org/wiki/Big_O_n...mal_definition. I figured it might be better to show the page first and then let you ask any specifics if you choose to at your own pleasure. 


#3
Mar1312, 07:26 AM

P: 625

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 onevariable. 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
Mar1312, 07:59 AM

P: 424

Question on BigO notation
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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