Big-O Notation Question: Polynomial with Variable Coefficients

[mnb96]

Hello,

I have a polynomial having the form:

$$\beta x^5 + \beta^2 x^7 + \beta^3 x^9 + \ldots = \sum_{n=1}^{+\infty}\beta^n x^{2n+3}$$

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 $O(x^5)$.

 

[chiro]

Hey mnb96.

My answer was going to be based on this page: http://en.wikipedia.org/wiki/Big_O_notation#Formal_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.

 

[mnb96]

First of all I forgot to mention that $x\geq 0$ and $\beta \in \mathbb{R}$.

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: $$f(x)=\beta x^5 + \beta^2 x^7 + \beta^3 x^9\ldots$$

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

By considering β variable, and x constant we have $f(\beta) \in O(\beta)$.

Now what?

 

[rasmhop]

So you want x->0 and $\beta \to 0$ right?

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