# Speed of convergence

• March 1st 2009, 10:01 AM
HTale
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 $\delta > 0$ 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.