Series

SERIES. The following notations will be used freely through out this article. The modulus of x, denoted by means the solute numerical value of x, when x is real; and when x is complex of the form it means the positive square root of The expression f(x)—>l, x-->a, means that f(x) tends to a limit 1, as x tends to a in any manner; if we write x-->a-{- or x—>a-- , this means that as x tends to a it always remains greater than a or less than a respectively. If n is a positive integer f(n).-->l means that f(n) tends to the limit 1 as a tends to infinity. The symbol 0(x) means " of the order of x." The symbol means "greater than or equal to" and the symbol means "less than or equal to." Logarithms, wherever mentioned, are to the base e.

A set of numbers to, • • • corresponding unequivocally to the set of positive integers 1, 2, 3, • • is called a sequence (see NUMBER SEQUENCES). The terms of a sequence, written in order, with the sign of addition between every adjacent two, thus -Eu2+ u3+ • • • , are said to form a series. If the sum of the first n terms of the series, • • • is denoted by s„, the study of the series is the study of the sequence 52, • • ' . If the number of terms is unlimited the series is said to be an infinite series. The series itself may be denoted by /u„. If s„ tends to a finite limit S, as n tends to infinity, the series is said to be convergent and S is said to be its sum. Such a series is often said to converge to S. If s„ tends to infinity, positive or negative, the series is said to be divergent and to diverge to +co or to —00 , as the case may be. When the symbol 00 is written alone, it denotes +60 . If s„ oscillates between two numbers a and b, the series is said to oscillate finitely. If s„ may assume posi tive or negative values, increasing numerically without limit, the series is said to oscillate infinitely. In modern writings all series which are not convergent are usually called divergent, the distinction being specified if necessary.

The chief problem connected with an infinite series is to dis cover whether it is convergent. The formal necessary and suffi cient test for convergence is that, e being any arbitrary small positive number, it is possible to find an integer v, such that < E, provided only that v and v. As, however, it is generally impossible to find any compact expression for s„, this test cannot be applied directly. We can, however, draw some use

ful conclusions from it. For example we must have and if u„-->o and never increases we must have The first shows at once that a series in which u„ is x"sin nO, cannot be convergent unless Ix' < I. The second shows that the series Ei/n is divergent. Neither condition is at all sufficient. The series i/n log n, for example, is divergent, although Since it is generally impossible to appeal directly to tests have to be con structed which depend on It may be said at once that com pletely general tests, both necessary and sufficient, have not been nor are likely to be found. Series can be constructed to defy any test, however delicate. But tests exist which are adequate to deal with nearly all such series as are likely to occur in applications.

Series of Positive Terms. The Ratio Tests.

There is an important difference between series in which all the terms, at least after some stage, are all of one sign and those in which the terms persist in being of variable sign. If the terms are all positive the terms of the sequence s„ are steadily increasing and it is not only a necessary but also a sufficient condition for s„ to tend to a limit that a number A can be found, such that A, for all values of n. If no such number can be found, then s„---->00 A series of positive terms cannot oscillate. If the terms are of variable sign, the condition A is necessary but is sufficient only to exclude infinite oscillation.

One of the simplest tests applicable to series of positive terms is that, if and are two series of positive-terms, then the convergence or divergence of one implies the convergence or divergence of the other, if tends to a finite limit, or if two positive numbers A and B can be found, such that A> B. In this way, convergence of Eu„ can be deduced from that of a series /v„, whose convergence or divergence is readily established. Thus is convergent if fn(n-1- Oh- is convergent. But if N.= n(n+ s„= i/(n+i), and s„-->i, so that /v„ is convergent and therefore is convergent. If u„/v„-->o, then lu„ is convergent, if /v„ is convergent, but both series may be divergent. If and v„ may have variable signs, the existence of a limit for is no indication that the convergence of Ev„ implies that of /u„.