Series

Series, in Algebra, is the sum of a set of terms formed according to some definite law. For example, let $n$ be any integer, and $\backslash phi(n)$ a definite function of $n$. Then, by giving $n$ the successive values 1, 2, 3, &c., and forming the corresponding functions $\backslash phi(1)$, $\backslash phi(2)$, &c., we are able to construct the series $S\; =\; \backslash phi(1)\; +\; \backslash phi(2)\; +\; \backslash dots\; +\; \backslash phi(n)$, where $n$ is the highest value of $n$ that is to be involved. If $\backslash phi(n)$ is simply a multiple of $n$, we get an Arithmetical Progression (q.v.), viz. $a\; +\; 2a\; +\; 3a\; +\; \backslash dots$. Again, if $\backslash phi(n)$ is of the form $a^n$, we get a Geometrical Progression (q.v.), viz. $a\; +\; a^2\; +\; a^3\; +\; \backslash dots$. These simplest cases of series are considered under their special headings, and shall not be again referred to except by way of illustration.

It is evident that if a finite number of terms be taken, and if no term has an infinite value, the series itself will have a finite and determinate value. We may suppose, however, that no limit is to be assigned to the number of terms that are to be taken—in other words, that the highest value ($n$) of $n$ is to be larger than any assignable quantity. We thus get a series with an infinite number of terms. But it does not follow that such an Infinite Series, as it is called, has necessarily an infinite value. Consider, for example, the Geometrical Series $1\; +\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{4\}\; +\; \backslash frac\{1\}\{8\}\; +\; \backslash frac\{1\}\{16\}\; +\; \backslash dots$ to infinity. Draw a line ABC equal in length to two

A ————— B ————— D ————— E ————— F ————— C

units. AB (= 1) will represent the first term of the series; the second term may be represented by BD, the half of BC; the third by DE, the half of DC; and so on indefinitely. It is evident that, however far we may go, we shall always fall short of C by an amount equal to the last bit added on. Thus

But by taking $n$ large enough we may make $1/2^n$ as small as we please. Hence the value of the Infinite Series is 2.

It will be seen that the terms in this series approach zero indefinitely, while the sum approaches a definite limit. Any series in which the latter condition is satisfied is called a Convergent Series. In all convergent series the former condition just stated must also be satisfied. But it does not follow that a series whose successive terms approach zero indefinitely is necessarily convergent. For example, the series $1\; +\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{3\}\; +\; \backslash frac\{1\}\{4\}\; +$ $\backslash frac\{1\}\{2\}\; +\; \backslash dots$ to infinity has not a finite value, is not convergent, although its infinite term is zero. Such a series is divergent, and cannot be summed to infinity. To prove this throw the series into groups of terms, the first group being the first term, the second the next two, the third the next four, the fourth the next eight, and so on. Thus the fifth group will consist of sixteen terms, beginning with $\backslash frac\{1\}\{16\}$ and ending with $\backslash frac\{1\}\{31\}$. Each of these fractions is greater than $\backslash frac\{1\}\{32\}$ or $1/2^5$; so that their sum is greater than sixteen times this quantity or $2^4/2^5$ or $\backslash frac\{1\}\{2\}$. Hence, if we go as far as $m$ groups, the series will be greater than $1\; +\; \backslash frac\{1\}\{2\}m$. Thus by taking $m$ large enough we can make the sum as large as we please. The series is divergent and cannot be summed. We may, however, by simply changing the algebraic sign of every alternate term, obtain a series which is convergent—viz. $1\; -\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{3\}\; -\; \backslash frac\{1\}\{4\}\; +\; \backslash dots$ to infinity. That this series has a finite sum may be made evident graphically thus: Take AB equal to unity; this is the first

term. Move back half-way to C; this gives the second term minus one-half. Move forward to D, where CD is one-third; then back to E, where DE is one-fourth; and so on indefinitely. It is clear that we shall ultimately oscillate through diminishing ranges about some point between C and B; so that the sum of this series is less than 1 but greater than $\backslash frac\{1\}\{2\}$. The series, in fact, is the Napierian logarithm of 2, and has the value .69315... A series like that just given, which is convergent only when the signs of the successive terms differ according to some definite rule, is usually called semi-convergent. A series which converges when all its terms have the same sign is said to be absolutely convergent. Sir G. B. Stokes long ago distinguished them as accidentally and essentially convergent, a terminology which seems in many respects superior to that in common use.

It is important to have a test of convergency; and the most useful test is to take the ratio of two consecutive terms, and consider what value this ratio approaches as we take the terms higher and higher. This ratio is called the ratio of convergency; if it is ultimately less than unity the series is convergent; if greater than unity, divergent. This test, however, gives no information when the ratio is ultimately unity. As an example, consider the exponential series:

Here the ratio of the $(n+1)$ to the $n$th term is $x/(n+1)$, which is ultimately zero, since whatever value $x$ may have $n$ can be taken as large as we please, so that the ratio may be made smaller than any assignable quantity. As is well known, the value of this series is $e^x$, where $e$ has the value 2.71828... (see LOGARITHM). Convergent series are of indispensable service in the calculation of logarithms and trigonometrical functions and in many important physical applications. Not a few of their properties were consequently known to the earlier analysts; but it is to Cauchy (1827) that we owe the foundation and partial development of the modern theory of convergence. Dirichlet, Abel, Gauss, De Morgan, Bertrand, Kummer, Du Bois-Reymond, and others have ably supplemented Cauchy's work. A very complete introduction to the whole subject is given in Chrystal's Algebra (vol. ii.). There also will be found a discussion of certain parts of the subject which we can only name, such as oscillating series, double series, infinite products, reversion of series, and the like.

See CIRCLE and TRIGONOMETRY for some particular cases of series.