NUMBER SEQUENCES. A set of numbers is said to form an ordered set if, in addition to the definition of the elements in the set, there is also given a means by which relative rank may be assigned to any two numbers of the set so that of two such num bers A and B one may say either that A precedes (is of lower rank than) B or that A follows (is of higher rank than) B. For example, the set 1, 3, 4 is an ordered set. The order relation so defined is analogous to that of points on a directed line and hence the order itself may be characterized as linear. By a part of a sequence we mean an ordered subset of its elements such that every two ele ments in the subset have the same relative rank in it as they have in the whole set. Various types of ordered sets are known as sequences. Any ordered set containing only a finite number of elements is called a finite sequence. In this case the order may be defined by a law, as when we speak of the even positive integers less than ioo taken in order of magnitude ; or it may be given by an exhibition of the objects in a given order, as when one writes the sequence 9, 7, 3, 6, I, 8, so. A simply infinite sequence, or a simple sequence, is an ordered set which contains no element of higher rank than all the others, while every part of it which con tains an element of higher rank than all the other elements in that part is a finite sequence. Thus the positive integers in the order of their magnitude form a simply infinite sequence. But if they are taken in the order I, 3, 5, 7, • • • , 4, • • • , they form not a simply infinite sequence but a combination of two such sequences. In a simply infinite sequence the elements are arranged in a countable order, that is, so that there is a first ele ment, a second, a third, and so on. Thus a simply infinite sequence may be denoted by the symbols ul, u2, u3, . . . , un, . . . , where it is understood that there is no last element in the sequence. The rational numbers between o and 1, taken in order of magnitude, af ford an example of an ordered set which cannot be denoted by such a sequence of symbols; in fact, there is no element in this set which has a next following element in the set. This set may, how
ever, be re-ordered so as to afford an example of a simple sequence. It is necessary that the order relation in an infinite sequence be prescribed by some norm or rule; it cannot be exhibited explicitly by a given arrangement as in the case of a finite sequence. In the course of the article other types of infinite sequences will appear.
The principle of mathematical induction may be given its most characteristic formulation with respect to simply infinite se quences. If in the case of a given simply infinite sequence it be true (I) that, if an element of the sequence possesses a given prop erty P, the next following element also possesses the property P; (2) that the first element in the sequence possesses this property P; then it is true that every element in the sequence possesses the property P. For instance, if we wish to show that the numbers in the sequence I, 1+3, 1+3+5, 1+3+5+7, . . . are the square numbers 42, . . . in order, we observe that the first term in the sequence has the required property and we prove that if one term has the property the next one also has it, and then we conclude to the truth of the general proposition by aid of the principle as formulated.
A sequence of numbers is said to form a harmonic progression if the reciprocals of its terms in order constitute an arithmetic pro In his papers of 1925 in French, he applied the same formula to find the number of representations as a sum 2, 4, 6, 8, io squares, and to deduce all known and certain new relations between the numbers of classes of positive binary quadratic forms of various determinants.