Sequences of Numbers

by George E. Hrabovsky, President, MAST

Where We Have Been

We have been continuing our focus on the foundations of mathematics. We have discussed the basic ideas of set theory and logic, proof techniques, quantifiers, and have begun to develop the number system. Last column we discussed the properties of irrational numbers.

Where Will We Go in This Column

In this column we will lay the foundations for discussing the real numbers.

Sequences

Any list of elements that has a definite order, that is there is a first element, a second element, and so on, is a sequence of elements or just a sequence. Each element of the sequence is called a term of the sequence, or just a term. The number of terms in a sequence is the order of the sequence. If we use the letter s as the symbol for a term, we can list the terms by adding a subscript with the order of the term

{s _ 1, s _ 2, s _ 3, ..., s _ n},

Thus s _ 1 is the first term, s _ 2 the second, and so on. We can construct a symbolic representation of a sequence by listing the order of a term along with the term, thus the nth term of a sequence would be

[(n, s _ n)] .

Since this pair of symbols has a definite order we will call it an ordered pair.
    Some sequences have a specific number of terms, these are called finite sequences. A sequence that is not finite is an infinite sequence. The set of elements from which the terms are drawn is called the domain of the sequence. The set of terms that actually appear in the sequence is called the range of the sequence.
    In mathematics the most useful sequences are sequences of numbers. For example, we could list the set of even natural numbers as a sequence,

{2, 4, 6, ..., 2 n, ...}

or

[(n, 2 n)] .

Such numerical sequences will be what we consider unless otherwise stated.

Representing Numbers as Sequences

Any system where numbers are represented symbolically is called a system of numeration. The symbols within such a system are called numerals. Any system where numbers are represented in such a way that the position of a numeral determines its order is called a positional system. The decimal numeration system is an example where we have ones, tens, hundreds, etc. In such a system we can treat each representation as a sequence. For example, if we have the number 1279, this is the sequence {1000,200,70,9}. A rational number can be represented this way,

1/3 = 0. Overscript[3333, _] = {0.3, 0.03, 0.003, 0.0003, ...} .

Here I used the convention of an overscore to indicate that the overscored sequence repeats itself endlessly.

The Cauchy Sequence

Any sequence whose higher order terms are smaller than previous terms is called a Cauchy sequence (after the famous mathematician Augustin Louis Cauchy, 1789-1857). It is easy to see that the above example of the sequence representing 1/3 is a Cauchy sequence, as each term of the sequence is progressively smaller than its predecessor.

Representing Irrational Numbers as Cauchy Sequences

In the last column ("Irrational Numbers") we discussed the irrationality of 2^(1/2). It turns out that the irrational answer is a sequence that is infinite. We can round this off to an answer that is close to the actual value (such an answer is called an approximation). An example of such an answer is 1.41421. We can see that this is a sequence, {1, 0.4, 0.01, 0.004, 0.0002, 0.00001}. This is a Cauchy sequence since successive terms are smaller.

•Suggested Practice Problems

1. What is the domain of a finite sequence?
2. Give three examples of numerical sequences.
3. Give three examples of numbers as sequences.
4. Determine if any of your numbers from problem 3 are Cauchy sequences.
5. Are numbers such as 23, 4, 0.5 Cauchy sequences? Why?
6. Is 1/3 a Cauchy sequence?

Converted by Mathematica  (January 15, 2003)