To keep track of position and time and quantities of different sorts, such as mass, volume, temperature, velocity,…, we use numbers:
- denotes the set of natural numbers
- denotes the set of integers ,
- denotes the set of rational numbers , where with .
A rational number has a decimal representation or expansion with a finite number of non-zero decimals or a neverending periodically repeating decimal representation. For example, and
A neverending decimal representation which is not periodic , is said to be an irrational number. The set of rational and irrational numbers with finite or infinite decimal representations forms the set of real numbers denoted by .
In constructive mathematics the decimals of decimal expansions, typically of roots to given equations, are computed successively by iterative methods such as fixed point iteration with Newton’s method as special case, which generate sequences of numbers as successively more accurate root approximations.
We say that a sequence of real numbers , also denoted or for short, is convergent if for a positive constant and a positive constant :
- for (1)
A convergent sequence of real numbers specifies a unique decimal expansion which defines a real number called the limit of the sequence, satisfying
- for (2)
To see this we note that for :
where we used the following formula for the sum of a finite geometric series with :
For example, if and , then it follows from (2) that the first decimals of are specified by the first decimals of , for .
We shall see that a convergent sequence is typically constructed by time stepping or fixed point iteration (with Newton’s method as an important special case) with each iteration determining a new decimal of the limit if (and binary digit if ) and as a root to an equation for a given function .
We shall see that for Newton’s method the number of correct decimals doubles with each iteration which gives a very fast convergence.
In short, we will construct convergent sequences by fixed point iteration with the decimals of the limit succesively being determined by the decimals of for increasing .
Example 1: Consider the sequence with , that is the sequence, . We see that
showing that the sequence satisfies (1) with and and thus converges, and we see that since .
Example 2: We shall discover (MST Chapter 8 and 190) that the sequence generated by the successive iteration for with , which is Newton’s method for computing a root of the equation , that is the sequence , converges and .
Example 3: Below we will meet a typical case of time stepping, where we will compare the result computed with time step and the result computed with half the time step , and find that with a constant independent of , which leads to a sequence satisfying (1) with , so that one binary digit of precision is gained in each step.
Remark: We will use the concepts of convergence and limit only with (1) as criterion, and not in the more general but more vague epsilon-delta form typically used in standard versions of Calculus (to define continuity and derivate in terms of limits):
- if for all there is such that for .