1.1 Basic Analysis

Analysis is the mathematical theory of infinite algorithms: evaluation of infinite sums, limits of arithmetic operations iterated infinitely many times, and so on. While no such algorithm can ever be implemented, some of its properties can be determined a priori without ever completing the calculation. What is more, a large finite number of arithmetic operations from a convergent infinite algorithm will produce a result close to the limit, whose properties will be similar to those of the limit. We will have a truncated infinite algorithm if we simply stop after a sufficiently large number of steps.

If a finite algorithm is a truncated infinite algorithm which has no limit, then the finite algorithm will be unstable, i. e., the result will vary greatly with differing truncations. On the other hand, truncating a convergent infinite process at any sufficiently large number of steps will produce essentially the same result. This is the idea behind Cauchy's criterion for a sequence { a( n)}: For each ? > 0 we can find a sufficiently large N