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in its relation to algebra and algebraical development. Definite integration also yields a new, and in some respects a more convenient form of the remainders of the theorems of Taylor and Maclaurin; and gives development of certain functions in periodic terms, which are not only curious, but are also important in the application of pure mathematics to questions of physics. The investigation of these theorems will occupy the present chapter; but previously to that inquiry it is desirable to give a succinct account of series.

178.] A series is a succession of terms, of which the number is infinite, placed one after another in a given order, and formed according to an uniform and determinate law, so that each term is a function of the place which it holds in the series. The function which expresses each in terms of its place gives the law of the series; and it is evident that the place of a term in a series cannot be changed without a change of law of the series. The following are the forms of series which we shall take; Uy+U9 + Uz + ...... +Um + Un+1+...;

(1) f(a) + f(a+1)+ f(a+2)+...+f(a+n-1) + f(a+n) + ...; () 1. 1 , 1

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the nth or the general term being respectively Uns f (a +n-1),

1: and consequently the law of the series is given, when these terms are expressed as functions of n.

For conciseness of notation we shall express these series severally by the symbols Xiun; f(x); No fla; which indicate the sum of a series of terms formed by assigning to x all integer values from the inferior limit to infinity. This notation is analogous to that of a definite integral.

A series is said to be convergent, when the sum of an infinite number of terms of it is a finite quantity; and this finite quantity is called the sum of the series.

A series is said to be divergent, when the sum of an infinite number of terms of it is infinite; in this case it is said that the series cannot be summed.

Thus if sn is the sum of the first n terms of the series given above, if the series is convergent, s, is finite, when n = 0. If

the series is divergent, sn = 0, when n = oo. When n = 0o, the sum of the series is denoted by s.

The sum of all the terms of a series after the nth is called the remainder of the series. Let it be denoted by R; then taking the form of the series given in (1), R = Un+1+Wn+2+...

(4) = S-S, It is to be observed that although the sum of a convergent series is a finite quantity, yet frequently that quantity cannot be determined in the form of a definite function. It may be capable of expression as a definite integral, which does not admit of evaluation; or it may be expressed as a function of certain letters contained in the series, as the notation of Art. 145 signifies : but in many cases it does not admit of more definite determination.

The remainder of a convergent series is infinitesimal, when n = 0; so that s, and sptr differ by an infinitesimal, which must be neglected when n = 00. Hence it follows that, when a series is convergent, not only must un be infinitesimal, and consequently be neglected, when n = 0, but also R, which is the sum of all these infinitesimals, must also be an infinitesimal, and be neglected when n = 0. Thus although the vanishing of each term, when n = 0, is true of all convergent series, yet this circumstance does not afford a sufficient test of convergence. It is a necessary characteristic, but is evidently not a sufficient criterion.

It will be observed that these definitions have respect to the terminal terms of the series; that is, those for which n = 0; thus although a series may begin divergently and continue divergently for a given number of terms, yet if it ultimately converges, it will satisfy the condition of convergence, and will be classed as a convergent series.

179.] Before we enter on the investigation of a strict criterion of convergence, it is worth while to examine two series of typical forms : viz., series in geometrical and in harmonical progression : for we shall hereby exemplify certain phenomena of series, the number of the terms of which is infinity; and we shall shew the necessity of either determining the convergence of a series when it and the function of which it is supposed to be the development are used as equivalents : or of ascertaining the remainder so that an exact equality may exist between the function and the series

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together with the remainder. Let us in the first place take the geometrical series a, ar, ara, ...... arh , ...

(6) of which arn-1 is the nth term. Then for the sum of the series to n terms we have

gono - 1 Sn = a

r_1 If r is less than 1, then, when n = oo, goue = 0, and

s = m; R = 5–s, = 2 = 0.
and consequently the series is convergent.
If r =l,

704 -1 0 .
Sn = a -1 = 7, if r = 1;

= anrn-1 = an, if r = 1.

.. s = 0, when n = 0; so that the series is divergent.

If r is greater than 1, then, when n = oc, gote = oo, and the series is also divergent.

Thus a series in geometrical progression is convergent for all values of r, the common ratio, less than 1; and is divergent for all values of r not less than 1.

If however we had without examination assumed the series in geometrical progression to be convergent for all values of r, and had omitted the remainder, then we should have had

s= o = a {1+r+ pod +93 +...}. (9) Now in this series let us suppose a = 1, r = 2; then

-1=1+2+4+8+ ... a result which is evidently absurd; and consequently proves that the left-hand member of (9) cannot be employed equivalently for the right-hand member, unless r is less than 1. If however the remainder is introduced we have an exact equality; and then

po -1 a{1+r+p2 + ... +yon-1} = a ; and if a = 1, r = 2, as before,

-1 = 1+2+4+...+2"-1-2", which is a correct result. This is an instance shewing the

(10) necessity of determining the remainder in the theorems of Maclaurin and Taylor; and shews how erroneous results may be when these theorems are applied, and either the resulting series are not convergent, or the remainders are not determined. It is for this reason, that in the former parts of our treatise, we have never employed series as the basis or subject-matter of argument, until the convergence of them has been demonstrated.

Again, let us consider the following series in harmonical progression.

1 1 1
- 5+ätāt...tata

+...; (11) this may be expressed in the following form ;

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now each group within brackets is evidently greater than; and as the number of such groups is infinite, the sum of the series is infinite, and consequently the series is divergent. This is an example in which Un = 0, when n = oo, and yet the series is not convergent.

This series may also be expressed in the following form, from which a remarkable result arises, and shews the divergence of the series.

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= 1+s: Thus 1 is an infinitesimal in comparison of s; in other words 8 = 00, s being that quantity of which a finite number is the infinitesimal increment.

180.] We now come to the investigation of general theorems for the determination of convergence and divergence of series. I shall arrange them as far as possible in order of simplicity, for that is the order in which they are most conveniently applied to any particular series.

If the terms of a series decrease in absolute magnitude, and are, or ultimately become, alternately positive and negative, the series is convergent. Let the series be Un — Uz + Ug — U4+...;

(12) the terms of which become less and less. Then the series may be expressed in the two following forms

(Uz — U2) + (Uz — Ux)+ (U7 – Us)+...;

Uy-(Uz — Uz) – (U4 - Uz) — ...; every quantity within brackets being positive. Whence it is evident that the sum of the series is greater than U, — Ug, and is less than Uy; consequently the series is convergent.

Ex. 1. Thus the series 2 -1is convergent.



The series is

+Ba+Bat...; which may be expressed 1 1

in the form 1-+ - +..which is by the preceding theorem greater than 1, and less than

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181.] The series U + Uz + Ug + ..., all of whose terms are positive, is convergent or divergent according as the ratio of Uz+1 to uz, when x = oo, is less or greater than unity.

Let r be the value of 4x+1, when x = 00; and firstly let r be less than 1; let p be greater than r and less than 1; then for some determinate value of x, and for all other greater values, the ratio of Uz+1 to uz is less than p; so that if x is that value,

Ux+1 < p Ugg Ux+2 < p Uz+1) Uzt3 < pUx+3) ...... ; consequently

Uz +Wx+1 + Uz++ ... <ux {1+p+p+ ... }


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