Rates of convergence

Given a sequence of rational numbers that sequence can either converge or diverge. If it converges to some point in the number line, then it does so at some rate. It can either converge to a rational number or to an irrational number. If it converges to some rational number then it is equal to some rational number plus or minus some infinitesimally small sequence that converges to zero. It is well known, for example, that the reciprocal 1/n converges to zero, so a number like 5+1/n converges to 5. The reciprocal of another function like 1/n^2 converges to zero quickly, so 5+1/n^5 converges to 5 quicker then 5+1/n. This difference is expressed in rates of convergence.

If we have some sequence that converges to an irrational number, then the same principle applies analogously. Consider the Leibniz formula, it converges to pi, but it does so extremely slowly. In order to get an accurate approximation of pi, it is not a very efficient sequence to use. The Ramanujan formula and Chudnovsky formula converge much faster. This shows that irrational numbers can be approximated to a greater or better extent by different sequences. An irrational number can then be defined be the most convergent sequence that is currently available. This is similar to defining them by equivalence classes, except this allows you to actually compute rational approximations.