Series

No part of Mathematics suffers more from

the triviality of its initial presentation to beginners than the great subject of series. Two minor examples of series, namely arithmetic and geometric series, are considered; these examples are important because they are the simplest examples of an important general theory. But the general ideas are never disclosed; and thus the examples, which exemplify nothing, are reduced to silly trivialities.

The general mathematical idea of a series is that of a set of things ranged in order, that is, in sequence; This meaning is accurately represented in the common use of the term. Consider for example, the series of English Prime Ministers during the nineteenth century, arranged in the order of their first tenure of that office within the century. The series commences with William Pitt, and ends with

Lord Rosebery, who, appropriately enough, is the biographer of the first member. We

might have considered other serial orders for the arrangement of these men; for example, according to their height or their weight. These other suggested orders strike us as trivial in connection with Prime Ministers, and would not naturally occur to the mind; but abstractly they are just as good orders as any other. When one order among terms is very much more important or more obvious than other orders, it is often spoken of as the order of those terms. Thus the order of the integers would always be taken to mean their order as arranged in order of magnitude. But of course there is an indefinite number of other ways of arranging them. When the number of things considered is finite, the number of ways of arranging them in order is called the number of their permutations. The number of permutations of a set of $n$ things, where $n$ is some finite integer, is $n \times (n - 1) \times (n - 2) \times (n - 3) \times \dots \times 4 \times 3 \times 2 \times 1,$ that is to say, it is the product of the first $n$ integers; this product is so important in mathematics that a special symbolism, is used for it, and it is always written ` $n!$ .' Thus, $2! = 2 \times 1 = 2$ , and $3! = 3 \times 2 \times 1 = 6$ , and $4! = 4 \times 3 \times 2 \times 1 = 24$ , and $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ . As $n$ increases, the value of $n!$ increases very quickly; thus $100!$ is a hundred times as large as $99!$ .

It is easy to verify in the case of small values of $n$ that $n!$ is the number of ways of arranging $n$ things in order. Thus consider two things $a$ and $b$ ; these are capable of the two orders $a b$ and $b a$ , and $2! = 2$ .

Again, take three things $a$ , $b$ , and $c$ ; these are capable of the six orders, $a b c$ , $a c b$ , $b a c$ , $b c a$ , $c a b$ , $c b a$ , and $3! = 6$ . Similarly for the twenty-four orders in which four things $a$ , $b$ , $c$ , and $d$ , can be arranged.

When we come to the infinite sets of things–-like

the sets of all the integers, or all the fractions, or all the real numbers for instance–-we come at once upon the complications of the theory of order-types. This subject was touched upon in VI. in considering the possible orders of the integers, and of the fractions, and of the real numbers. The whole question of order-types forms a comparatively new branch of mathematics of great importance. We shall not consider it any further. All the infinite series which we consider now are of the same order-type as the integers arranged in ascending