What's a sequence and why should you care?

Definition 1   A sequence of real numbers is a function from the natural numbers to the reals.We typically use subscripts rather than functional notation using parentheses to indicate this so that the sequence $a:\ensuremath{\mathbb{N} }\to \ensuremath{\mathbb{R} }$ is usually written as $$a_0 , a_1 , \ldots ,a_n \ldots$$

Sequences are important in approximation: the usual representation of real numbers using decimals is in fact the process of giving a sequence of rational numbers approximating the real number in question successively better as more decimal places are given. Most of the important functions from the reals to the reals which we use are actually only able to be calculated approximately. Series representations (based on sequences of real numbers) provide the means to get arbitrarily good approximations. Sequences were also used by Cauchy to construct the real numbers from the rationals.
 

1999-11-14