# Orders of convergence

• Oct 3rd 2012, 01:24 AM
alanaj5
Orders of convergence
Hi Guys,

I have a small question on orders of convergence.
I'm happy with the idea that for $N\in\mathbb{N}$ if,
$\text{error} \le C_1N^{-\alpha}+C_2N^{-\beta}$, where $\alpha<\beta$, $\alpha, \beta \in\mathbb{R}$, $\alpha,\beta>0$, $C_1, C_2 \in\mathbb{R}$ are constants
$\Rightarrow \text{error} \le (C_1+C_2)N^{-\alpha}$

or

$\text{error} \le C_1N^{\alpha}+C_2N^{\beta}$, where $\alpha<\beta$, $\alpha, \beta \in\mathbb{R}$, $\alpha,\beta>0$, $C_1, C_2 \in\mathbb{R}$ are constants
$\Rightarrow \text{error} \le (C_1+C_2)N^{\beta}$

but for
$\text{error} \le C_1N^{-\alpha}+C_2N^{\beta}$, where $\alpha, \beta \in\mathbb{R}$, $\alpha,\beta>0$, $C_1, C_2 \in\mathbb{R}$ are constants.
is this the right outcome
$\text{error} \le (C_1+C_2)N^{\beta}$ ?
Other than when 0 < N < 1 is there a time when $N^{-\alpha}$ would win?
Thanks :-)
• Oct 3rd 2012, 01:50 AM
alanaj5
Re: Orders of convergence
I think I might have just figured out what I needed to know. I think the size of the coefficients $C_1$ and $C_2$ play an important role here as to which rate dominates the other. Please let me know if you think I'm wrong or I should be considering other factors as well.

Thanks.