Hi Guys,

I have a small question on orders of convergence.

I'm happy with the idea that for $\displaystyle $N\in\mathbb{N}$$ if,

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

$\displaystyle $\Rightarrow \text{error} \le (C_1+C_2)N^{-\alpha}$$

or

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

$\displaystyle $\Rightarrow \text{error} \le (C_1+C_2)N^{\beta}$$

but for

$\displaystyle $\text{error} \le C_1N^{-\alpha}+C_2N^{\beta}$$, where $\displaystyle $\alpha, \beta \in\mathbb{R}$$, $\displaystyle $\alpha,\beta>0$$,$\displaystyle $C_1, C_2 \in\mathbb{R}$ $ are constants.

is this the right outcome

$\displaystyle $\text{error} \le (C_1+C_2)N^{\beta}$$ ?

Other than when 0 < N < 1 is there a time when $\displaystyle $N^{-\alpha}$$ would win?

Thanks :-)