Register to reply

Question on Big-O notation

by mnb96
Tags: bigo, notation
Share this thread:
Mar13-12, 05:16 AM
P: 626

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].
Phys.Org News Partner Mathematics news on
Math journal puts Rauzy fractcal image on the cover
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
Mar13-12, 05:34 AM
P: 4,577
Hey mnb96.

My answer was going to be based on this page:

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.
Mar13-12, 07:26 AM
P: 626
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?

Mar13-12, 07:59 AM
P: 424
Question on Big-O 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.

Register to reply

Related Discussions
Notation question. General Math 2
Question about notation Set Theory, Logic, Probability, Statistics 3
Notation question General Math 2
Notation question Set Theory, Logic, Probability, Statistics 1
Notation question General Math 0