Number Sequences

Other Types of Infinite Sequences.

A simply infinite se quence is said to be of type w; the distinguishing qualities of these sequences lie in their possessing a first term, a term following any given term but no last term, while the terms preceding any given term constitute a finite sequence. The symbol *w is used to denote what may be called a reversed simply infinite sequence. It has no first term but does have a last term, while every term but the last one is followed by a next term and all the terms following a given term constitute a finite sequence. A sequence of type w is called an ascending sequence while one of type *co is called a descending sequence. As the positive integers 1, 2, 3, . . . in order of mag nitude form a typical ascending sequence so the negative integers in order of magnitude . . . , —, 3 — 2, — I form a typical de scending sequence. The sequence . . . , —3, -2, -I, 0, I, 2, 3, . . .

of positive and negative numbers and o, taken in order of magni tude, form a sequence of type *co-f-co, characterized by its posses sion of the following properties : there is no first term and no last term, there is a term next following any given term, and the elements following one and preceding another given element form a finite sequence.

In the development of the general theory of functions it has become necessary to consider many other types of sequences as well as order types of a more general character than those for which the term sequence is usually employed ; on this account a general theory of sequences and order types has sprung up— usually developed both in treatises on functions of a real variable and in those on the theory of sets of points (see POINT SETS). A few of those usually called sequences may be briefly described. A sequence of type co followed by another of type co is said to form a sequence of type C0.2. A typical instance is the following : I, 3, 5, 7, • • • , 2, 4, 6, 8, . . . .

If n such simple sequences are taken in a given order the resulting sequence is said to be of type w•n. A single sequence of type w will then be noted by co.'. If any sequence of type co•n is followed by a finite sequence containing m elements, the sequence so formed is said to be of type CO•11+M.

Let S2, . . . denote a simply infinite sequence each element of which is itself a simply infinite sequence. The sequence thus formed is said to be of type O. This process of forming sequences of sequences may be continued ; it has given rise to a certain class of so-called transfinite numbers. The theory of these transfinite numbers, and of the sequences which underlie them, has been extensively developed ; and it has given rise to important analyses of the logical processes involved in defining them and reasoning about them, processes usually investigated in the theory of sets of points, since these sequences may be represented to the mind by means of sets of points on a line.

Let us illustrate the last remark by exhibiting a set of points of type First define any simply infinite sequence of non-over lapping intervals on a given line, as, for instance, the intervals from o to 1, from I to II, from II to It and so on without end, each interval after the first having half the length of the preceding one.

On each of the intervals a set of points forming a sequence of type co is to be defined. We might, for instance, define the points on a given interval / as follows. Let the first one be the midpoint of I, the second be the midpoint of the part of / to the right of and in general let the ith one Pi be the midpoint of the part of I to the right of Then on / we have a sequence Pi, P2, . . . of points of type w . When this is done for each interval I of the set of intervals and the resulting points are contemplated in their order from left to right on the line, we have a sequence of points of type co'.

A typical arrangement of a sequence of type is the following: the elements in each line forming a simply infinite sequence and the elements in any line preceding all the elements in each follow ing line. Such a sequence is often called a double sequence. It is the type which underlies the general theory of double series. A fundamental problem concerning double sequences is that of the existence and equality of the limits the last two denoting limits of limiting values while the first denotes the limit as m and n become infinite independently. If the first limit exists and is a then each of the other limits exists and is a. But one or both of the latter limits may exist while the first does not, nor is it necessary that the two latter limits (when they exist) shall be equal.

That the same set of objects may be contemplated in sequences of various types may be indicated by observing that the elements in a double sequence, or sequence of type are capable of an arrangement into a sequence of type o.). Thus, in the case of the foregoing array we may arrange the elements in the order of the finite diagonal sets and obtain the following sequence of type co: a11, a12, a21, a13, a22, a31, a14, 0-23, • • • • Recurrent Sequences of Integers.—In the theory of ordinary integral numbers certain simply infinite sequences of integers have been found to play an important role (Quart. Journ. Math., 48 [192o] : 342-372). These integers satisfy recurrence relations of the form set o, 1, 2, ... p— I, is periodic and (when certain exceptional cases are removed) the period is a factor of T. This theorem may be extended to the case of a general modulus m. In this connection several criteria have been obtained for recognizing certain large prime numbers as prime, the following being one of the simplest. A necessary and sufficient condition that 2" — I shall be a prime is that shall be divisible by 2" — I, where a and b are any integers such that is a prime number of which I is a quadratic non-residue. By means of such theorems several large numbers have been shown to be prime, including the following: See E. W. Hobson, Theory of Functions of a Real Variable, vol. i. (3rd ed., 1927) ; and Encyclopedie des Sciences Mathematiques, tome i., vol. i. (R. D. CA.)