Sequences, Series, and Function Approximation

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 approximation the real number in question successively better as more decimal places are given. These decimal approximation sequences are actually rather special: successive decimal approximations never get smaller (so the sequence is monotone nondecreasing) and two approximations which agree to the kth decimal place differ by at most 10-k (so the sequence is a Cauchy sequence: to make two values in the sequence close to each other all you need to do is take them far enough out in the sequence).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.