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 \delta > 0 and a sequence (p_n/q_n) of rational numbers such that each p_n, q_n \in \mathbb{Z}, with p_n/q_n \neq \beta and

\left|\beta - \frac{p_n}{q_n}\right|< \frac{1}{q_n^{1+\delta}} for n=1,2,\ldots

then \beta is irrational.

That is, why would one sequence be a better approximator for the irrationality of \beta than another.

Thanks in advance guys.