XXVI

LAW OF DISTRIBUTION FOR SERIES

127

successively three and four terms of each series, we see that the sum of (1) lies between 583 and 833; whereas the sum of (2) lies between 926 and 1.176.

Addition of two infinite series. If Eun and Evn be both convergent, and converge to the values S and T respectively, then Σ(un + vn) is convergent and converges to the value S + T.

We may, to secure complete generality, suppose un and vʼn to be complex quantities. Let S,, Tn, Un represent the sums of n terms of Zun, Σvn, Z(un + vn) respectively; then we have, however great n may be, Un = Sn + Tn. Hence, when n =

LU2 = LS2 + LT, which proves the proposition.

§ 14.] Law of Distribution.-The application of the law of distribution will be indicated by the following theorems :

If a be any finite quantity, and Zun converge to the value S, then Zaun converges to aS.

The proof of this is so simple that it may be left to the reader.

If Eun and Evn converge to the values S and T respectively, and at least one of the two series be absolutely convergent, then the series U ̧v1 + (U ̧v ̧ + U ̧v¡) + + (U ̧Vn + UşVn−1+ + Unv1) + . (1) converges to the value ST.*

...

Let Sn, Tn, Un denote the sums of n terms of Eun, Evn, Σ(U2Vn + UqVn−1 +. .+unv) respectively; and let us suppose that Zun is absolutely convergent.

where

We have

The original demonstration of this theorem given by Cauchy in his Analyse Algébrique required that both the series Zun, Zun be absolutely convergent. Abel's demonstration is subject to the same restriction. The more general form was given by Mertens, Crelle's Jour., lxxix. (1875). Abel had, however, proved a more general theorem (see § 20, Cor.), which cludes the result in question.

partly in

128

THEOREM OF CAUCHY AND MERTENS

CHAP.

2m say,

If therefore n be even, =

L = [U1⁄21⁄2m + U3(V2m+ V2m-1) + . . . + Um(V2m + ... + Vm+2)]

Ln = [UqV2m+1 + Uz(V2m+1+V2m) + ... + Um(V2m + . . . + Vm+3)] + [Um +1(V2m+1 + ... + V m + 2) + . . . + Ugm+1(V2m+1 + . . . + V2)] (4).

...

Now, since v is convergent, it is possible, by making m sufficiently great, to make each of the quantities mod vam, .. + Vam), mod v1⁄2m+1,

mod (Vam-12m),

mod (Vm+V2m+1),

mod (m+2 +

+ Vam+1) as small as we

please. Also, since mod T,, mod T2, mod T1,

mod Tn,

are all finite, and mod (T, - T.) < mod T, + mod T,, therefore

Em

are all finite. Hence, if em be a quantity which can be made as small as we please by sufficiently increasing m, and ẞ a certain finite quantity, we have, from (3) and (4), by chap. xii., § 11,

If, therefore, we make n infinite, and observe that, since Zum is absolutely convergent, mod u, + mod u, + . . . + mod un is finite, and L(mod um+1 + mod um+2+ + mod un) = 0, we have (seeing that Lem = 0) L mod L = 0. Hence LS2T2 = LUn that is, LU, ST.

Cauchy has shown that, if both the series involved be semiconvergent, the multiplication rule does not necessarily apply.

Suppose, for example, un=vn=(−1)-1/n. Then both Zu, and Zr, are semi-convergent series. The general term of (1) is

Now, since r(n−r+1) = {(n+1)2 − { } (n + 1) − r), therefore, for all values of r, r(n-r+1)<(n+1)2, except in the case where r=(n+1), and then there is equality. It follows that mod w2>n/}(n + 1) > 2/(1+1/n). The terms of Zw, are therefore ultimately numerically greater than a quantity which is infinitely nearly equal to 2. Hence Zwn cannot be a convergent series.

XXVI

RADIUS AND CIRCLE OF CONVERGENCY

129

SPECIAL DISCUSSION OF THE POWER SERIES Zanan.

§ 15.] As the series Zana" is of great importance in algebraical analysis we shall give a special discussion of its properties as regards both convergence and continuity. We may speak of it for shortness as the Power Series, and we shall consider both an and to be complex numbers; say an = rn(cos an + i sin an), x = p(cos + i sin 0), where in and an are functions of the integral variable n, but p and are independent of n.

§ 16.] Zanan is convergent if mod x < L {mod an/mod an+1} ·

For the series of moduli is Ernp", and this is convergent if L {p"+1rn+1/p"rn} <1; that is, if pL {n+1} <1; that is, if p<L {rn/fn+1} •

Three different cases arise according as L rn/n+1} is zero, a finite positive quantity R, or ∞o. In the first case, Zanan is not convergent for any value of x other than 0.

In the second case Zan" is convergent when the point representing x in Argand's Diagram lies within a circle whose centre is the origin and whose radius is R. This circle is called the Circle of Convergence for the power series in question; and R is called the Radius of Convergence. It should be observed that nothing is established for the case where the representative point lies on the circle of convergence.

Za/n is an example of this class of series; here R=1.

In the third case, Zan" is convergent for all values of x.
The exponential series Ex/n! is an example of this class of series.

§ 17.] If the series Zann be absolutely convergent when mod x = R', it will be absolutely convergent when mod x = R" <R'.

For, since Zana" is absolutely convergent, ErnR'" is convergent. Now, since R" <R', r,R"" <rR". Hence, by § 4, I, ErnR" is convergent; that is, Zanan is absolutely convergent when mod x = R".

§ 18.] Discontinuity and Infinitely slow Convergency. If the nth term of an infinite series be f(n,x), where f(n,x) is a single valued continuous function of x for all integral values of n, then

VOL. II

K

130

UNIFORM AND NON-UNIFORM CONVERGENCY

CHAP.

the infinite series Σf(n,r) will, if convergent, be a single valued finite function of x, say p(x). At first sight, it might be supposed that (2) must necessarily be continuous, seeing that each term of f(n,x) is so. Cauchy took this view; but, as Abel first pointed out, (x) is not necessarily continuous. No doubt Σf(n,x + h) and f(n,x), being each convergent, have each definite finite values, and therefore f(n,x + h) − f(n,x)} is convergent, and has a definite finite value; but this value is not necessarily zero for all values of x. that f(n,x) = x/(nx + 1) (nx − x + 1).

Suppose, for example,* Since f(n,x) = nx/(nx + 1) − this case, Sn = nx/(nx + 1).

(n − 1)x/ {n − 1x + 1}, we have, in Hence, provided x0, LS, = 1. If, however, x = 0 then S2 = 0, however great n may be. The function () is, therefore, in this case, discontinuous when x = 0.

The discontinuity of the above series is accompanied by another peculiarity which is often, although not always, associated with discontinuity. The Residue of the series, when

x0, is given by

R2 = 1-Sn = 1/(nx + 1).

Now, when has any given value, we can by making n large enough make 1/(nx + 1) smaller than any given positive quantity a. But, on the other hand, the smaller x is, the larger must we take n in order that 1/(nx + 1) may fall under a; and, in general, when x is variable, there is no finite upper limit for n, independent of x, say v, such that if n>v then Rn <a. When the residue has this peculiarity the series is said to be non-uniformly convergent; and, if for a particular value of x, such as x = 0, in the present example, the number of terms required to secure a given degree of approximation to the limit is infinite, the series is said. to Converge Infinitely Slowly.

We are thus led to the following important definition. If, for values of x within a given region in Argand's Diagram, we can for every value of a, however small mod a, assign for n an upper limit v, INDEPENDENT OF x, such that when n>v mod Rn <mod a,

Du Bois-Reymond.

XXVI DU BOIS-REYMOND'S THEOREM REGARDING CONTINUITY 131

then the series f(n,x) is said to be UNIFORMLY CONVERGENT within the region in question.*

It can be shown that, so long as Σf(n,x) converges uniformly, (x) cannot be discontinuous; but Du Bois-Reymond has shown by means of the example Σ {x/n(nx + 1) (nx − x + 1) − x2/(nx2 + 1) (nx3 − x + 1)} that infinitely slow convergence may not involve discontinuity. In point of fact, the sum of this last series is always zero, even when x=0; and yet, when x = = 0, the con

vergence is infinitely slow.

The object of the present paragraph has been, not to introduce the student to a discussion of exceptional cases in the functionality of infinite series, but to lead him to see the necessity of the demonstrations now to be given of the continuity of the Power Series in certain cases.

§ 19.] As regards the power series Zan" there are two cases of great practical importance-1st, when an is independent of x and we regard Zan as a function of 2, say (x); 2nd, when an is a function of n and y, say f(n,y), and x is regarded as constant, so that f(n,y)x" is a function of y, say +(y).

The points involved were first raised and discussed by Abel; but the following theorem, together with its elegant demonstration,† will give us at once all that is here required.

Let pn be independent of z, and w,(z) be a single valued function of n and z, finite for all values of n, however great, and finite and continuous as regards z from za to z = b, then, if Σμn be absolutely convergent, unwn(z)‡ is a continuous function of z from z = a = b.

to z =

Let S,()= w(≈) + μ‚w ̧(÷) + . . . + P(), and assume n to be positive for all values of n, which will not limit the

*The distinction here involved was first pointed out by Seidel, Abhl. d. Bayerischen Akad. d. Wiss., Bd. v. (1850). It has assumed great importance in the Theory of Functions developed by Weierstrass and his followers.

+ Both due to Du Bois-Reymond. See Math. Ann., iv. (1871). We have presented the original notation and phraseology as closely as possible.

Under the circumstances supposed, Zu,w() is, of course, convergent by § 4.