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 and a sequence of rational numbers such that each , with and
then is irrational.
That is, why would one sequence be a better approximator for the irrationality of than another.
Thanks in advance guys.