Physical Sciences and Mathematics Commons^™

Sequences, Series, And Function Approximation, Lawrence Stout

Lawrence N. Stout

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 …