am, x, &c.
The general term of a series is an algebraical for mula, which is usually a function of the quantities which are engaged in the series, and a general sym bol n, denoting the place of any term or its numeri cal order relatively to some term which is considered as the first term of the series, or the point of depart ure to which the places of all the other terms are re ferred. Thus in the last example, if am be consider ed as the first term, it appears that the number which is subtracted from m ill the exponent of a in each term is equal to the number of preceding terms, and therefore is one less than the number which denotes the place of the term itself; and the exponent of x in each term is the number which is subtracted from in in the exponent of a in the same term. Thus if n denote the numerical order of any term 'I', we have m—(n-1) n-1 T a It is obvious that the formula which expresses the general term of a series also expresses its law, and that the terms of the series may be found by substitut ing the successive integers 1, 2, 3, See. for n in this formula.
The terms of a series are generally connected to gether by the algebraical signs or —, and it is to the whole thus connected that the term series is ap plied.
A series generally arises from some analytical pro cess to which a function or formula has been submit ted, and this process is called development; the finite formula is said to be developed, and the resulting series is called its development.
There arc several methods of development adapted to the various forms which fractions may assume. The principal and the most general of these arc the method of indeterminate coeffeeients and the theorem of Under the latter is comprehended several .
methods which are sometimes enumerated separately; such as, the method of division, the binomial theo rem, Taylor's theorem, Maclaurin's theorem, &c. These methods the reader will find explained in the articles A soznits and Fsexioxs. Numerous appli cations of these processes will also he found in seve ral of our mathematical articles, as, TitIOONOMElitY, IsoosniTusis, &C.
The summation of a series is that process by which the algebraical sum of any number of its terms, or even of the whole series continued ad infinitum, may be found.
The summation of series is a subject w hich has en gaged the attention of the most eminent modern analysts, from the time of Wallis, who seems first to hate gisen the doctrine of series that consideration which its importance demands, to the present day. Any detailed account of these various methods of summation, which hase been proposed by modern mathematicians, would exceed those limits which we f.nd it expedient to impose on such a discussion in
this work. We shall therefore confine ourselves to a few examples, illustratise of some of the principal methods.
To determine the :nal of a Reri(4 of fractions, whose unmerators are in arithmeiral, aad whose denominators are in Afro:wirier!! progression; the number of /eons being supposed infinite.
Let the series be expressed this, if -1-d +2/1 id + a r r3 By the formulx established in article Asos.nits, for •umming a decreasing geometrical series continued rn we hate the following results: (Jr a' 4 - a r a'rl r —a' a'r a a r —a'a dr' r . = a'r It is evident that proposed series is the sum of the first members of these equations. if then we ex press this sum by S, we have S — or ± rl 5 4.
a' — ) ti(r r r r The series within the brackets, in the second mem ber of this equation, is also a geometrical series in which is the constant multiplier. Its sum continu• ed in infinitum, therefore — Hence we obtain r ar dr S = + a' (r — I) a' (r — Ifar (r — I) + dr or S ....--.
a' (r — The same method of summation may be applied to the case in which the numerators, instead of being in arithmetical progression, are composed of a constant quantity a, connected by addition, with terms com posed of a constant factor d, and ilie trian•lar num bers I, 3, 6, 10, Uc. Thus let the series 1,e,ft II + d a + lel a + e d s . _ + + + + a' a'r As before we obtain the following equations by the summation of geometrical series: a a a a a fir + + , -, + - - • - -- — a' a r a'r 4 • o'r- o'r — a el d d el d --7- + —, + • -F , i a r a'r- a'r • If 7' .--- fl 24 .,,d hd h I — --f- + -- -1-. - - - - u'rl a r--a r 74 4- — -;,1 i-- . I a'r* a' e..---a r2 4'l .1,1 a'r' a I' 1 ' As before the sum of the first members is S; so IliZtt. ICC haye ar d 2 5— , + 1 + +1 -4 cr — I) a' (r — , ) s r 7.2 r' ' S In the series which is within the brackets, the n „ merators are in arithmetical progression, and thaw: fore its HUM may be obtained by the general formula established in the preceding example. This w ill be elected hy making a — a' d • I, whence we obtai•I + 2 1 4 rs - + - + + r r2 rs (r--1,2 Hence we find ea- a' r—l) a By dividing the former series by r, we have I 2 1 4 r- ÷ + , + r r2 r- Ifence I 2.) 4- + ÷ ÷ 2 V + - - = 2 I 2 ,+ and so on.