
Speed of convergence
Hi guys,
I'm a little confused by the terminology "speed of convergence", particularly as applied to proving the irrationality of a number. What does it mean that one sequence converges "faster" than another? Specifically, in the use of the criterion for irrationality, namely,
If there is a $\displaystyle \delta > 0$ and a sequence $\displaystyle (p_n/q_n)$ of rational numbers such that each $\displaystyle p_n, q_n \in \mathbb{Z}$, with $\displaystyle p_n/q_n \neq \beta$ and
$\displaystyle \left\beta  \frac{p_n}{q_n}\right< \frac{1}{q_n^{1+\delta}}$ for $\displaystyle n=1,2,\ldots$
then $\displaystyle \beta$ is irrational.
That is, why would one sequence be a better approximator for the irrationality of $\displaystyle \beta$ than another.
Thanks in advance guys.