SERIES (a Latin word from serere, to join), a succession or sequence. In mathematics, the term is applied to a succession of arithmetical or algebraic quantities (see below); in geology it is synonymous with formation, and denotes a stage in the classification of strata, being superior to group (and consequently to bed, and zone or horizon) and inferior to system; in chemistry, the term is used particularly in the form homologous series, given to hydrocarbons of similar constitution and their derivatives which differ in empirical composition by a multiple of CH2, and in the form isologous series, applied to hydrocarbons and their derivatives which differ in empirical composition by a multiple of H2; it is also used in the form isomorphous series to denote elements related isomorphously. The word is also employed in zoological and botanical classification.
In mathematics a set of quantities, real or complex, arranged in order so that each quantity is definitely and uniquely deter-mined by its position, is said to form a series. Usually a series proceeds in one direction and the successive terms are denoted by ul, u2, ... u,,, ... ; we may, however, have a series proceeding in both directions, a back-and-forwards series, in which case the terms are denoted by
u_2, u—1, uo, U,, U2, ... U" ..
or its general term may depend on two integers positive or negative, and its general term may be denoted by u,,,, „; such a series is called a double series, and so on. The number of terms may be limited or unlimited, and we have two theories, (1) of finite series and (2) of infinite series. The first concerns itself mainly with the summation of a finite number of terms of the series; the notions of convergence and divergence present themselves in the theory of infinite series.
I. When we are given a series, it is supposed that we are given the law by which the general term is formed. The first few terms of a series afford no clue to the general term; the series of which the first four terms are 1, 2, 4, 8, may be the series of whioh the general term is 2”; it may equally well be the series of which the general term is *(n'+5n+6); in fact we can construct an infinite number of series of which the leading terms shall be any assigned quantities. The only case in which the series may be completely determined from its leading terms is that of a " recurring series." A recurring series is a series in which the consecutive terms, after the earlier ones, are connected by a linear relation; thus if we have a relation of the form
apur + ap_1ur+1 + a2_2ur+2+ . . . + T
the series is said to be a recurring series with a scale of relation
though taken from a genuine specimen; but little that can be called Ralline in character is observable therein. The same is to be said of an egg laid in captivity at Paris; but a specimen in Mr Walter's possession undeniably shows it (cf. Proc. Zool. Society, 1881, p. 2):
' A supposed fossil Cariama from the caves of Brazil, mentioned by Bonaparte (C.R. xliii. p. 779) and others, has since been shown by Reinhardt (Ibis, 1882, pp. 321-332) to rest upon the misinterpretation of certain bones, which the latter considers to have been those of a Rhea.
' Near Tucuman and Catamarca (Burmeister, Reise durch die La Plata Staaten, ii. p. 508).
observations on the birds of Paraguay (A puntamientos, No. 340), wherein he gave an account of it under the name of " Saria," which it bore among the Guaranis,—that of " Cariama " being applied to it by the Portuguese settlers, and both expressive of its ordinary cry.' It was not, however, until 1809 that this very remarkable form came to be autoptically described scientifically. This was done by the elder Geoffroy St-Hilaire (Ann. du museum, xiii. pp. 362-370, pl. 26), who had seen a specimen in the Lisbon museum; and, though knowing it had already been received into scientific nomenclature, he called it anew Microdactylus marcgravii. In 1811 J. K. W. Illiger, without having seen an example, renamed the genus Dicholophus—a term which has since been frequently applied to it—placing it in the curious congeries of forms having little affinity which he called Alectorides. In the course of his travels in Brazil (1815-1817), Prince Max of Wied met with this bird, and in 1823 there appeared from his pen N. Act. Acad. L.-C. nat. curiosorum, xi. pt. 2, pp. 341-350, tab. xlv.) a very good contribution to its history, embellished by a faithful life-sized figure of its head. The same year Temminck figured it in the Planches colorises (No. 237). It is not easy to say when any example of the bird first came under the eyes of British ornithologists; but in the Zoological Proceedings for
1836 (pp. 29-32) W. Martin described the visceral and osteological anatomy of one which had been received alive the preceding year.
The Seriema, owing to its long legs and neck, stands some two feet or more in height, and in menageries bears itself with a stately deportment. Its bright red beak, the bare bluish skin surrounding its large grey eyes, and the tufts of elongated feathers springing vertically from its lores, give it a pleasing and animated expression; but its plumage generally is of an inconspicuous ochreous grey above and dull white beneath,—the feathers of the upper parts, which on the neck and throat are long and loose, being barred by fine zigzag markings of dark brown, while those of the lower parts are more or less striped. The wing-quills are brownish black, banded with mottled white, and those of the tail, except the middle pair, which are wholly greyish brown, are banded with mottled white at the base and the tip, but dark brown for the rest of their length. The legs are red. The Seriema inhabits the campos or elevated open parts of Brazil, from the neighbourhood of Pernambuco to the Rio de la Plata, extending inland as far as Matto Grosso (long. 60°), and occurring also, though sparsely, in Paraguay. It lives in the high grass, running away in a stooping posture to avoid discovery on being approached, and taking flight only at the utmost need. Yet it builds' its nest in thick bushes or trees at about a man's height from the ground, therein laying two eggs, which Professor Burmeister likens to those of the Land-Rail in colour.' The young are hatched
ao -f aix + a2x2 + ... + a1,x'. It is clear that we can regard the series uo+uix+u2x2+...as the expansion in powers of x of an expression of the form
(bo+bix+ ... +-bn-ixP-1)/(ao+aix+ ... +avxp),
and by splitting this expression into partial fractions we can obtain the general term of the series. If we know that a series is a recurring series and know the number of terms in its scale of relation, we can determine this scale if we are given a sufficient number of terms of the series and obtain its general term. It follows that the general term of a recurring series is of the form L¢(n)an, where ¢(n) is a rational integral algebraic function of n, and a is independent of n. The series whose general term is of the form Kan+..(n), where 0(n) is a rational integral algebraic function of degree r, is a recurring series whose scale of relation is (I—ax) (i —x)**1, but the general term of this series may be obtained by another method. Suppose we have a series uo, u1, u2,... From this we can form a series vo, v2,.. where vn=un..i.1—un; from vo, v1, v2,... we similarly form another series and so on; we write vn-kun, and we suppose E to be an operation such that Eun=un+1 (the notation is that of the calculus of finite differences); the operations E and i +A are equivalent and hence the operations En and (i +0)n are equivalent, so that we obtain un =uo + n0uo +
A2uo+ ... This is true whatever the form of is.. When
is,. is of the form Kan+¢(n), where 0(n) is of degree r, 0''`°uo, Ci'42uo,
. form a geometrical progression, of which the common difference is a— 1, or vanish if the term Ka" is absent. In either case we readily obtain the expression for un.
2. The general problem of finite series is to find the sum of n terms of a series of which the law of formation is given. By finding the sum to n terms is meant finding some simple function of n, or a sum of a finite number of simple functions, the number being independent of n, which shall be equal to this sum. Such an expression cannot always be found even in the case of the simplest series. The sum of n terms of the arithmetic progression a, a+b, a+2b, ... is na+2n(n—i)b; the sum of n terms of the geometric progression a, ab, ab2, ... is a(i —b) ; yet we can find no simple expression to represent the sum of n terms of the harmonic progression
3. The only type of series that can be summed to n terms with complete generality is a recurring series. If we let Sn=uo+uix+
+un_,xn-1, where a5,. .. is a recurring series with a given scale of relation, for simplicity take it to be i+px+qx2, we shall have
Sn (i +px +qx2) = uo + (ul + puo) x + (pun_1 +qun_2) xn +qun_lxn+l,
If x had a value that made r+px+qx2 vanish, this method would fail, but we could find the sum in this case by finding the general term of the series. For particular cases of recurring series we may proceed somewhat differently. If the nth term is unxn we have from the equivalence of the operations E and i+A,
uix +u2x2 + . . . +unxn=xul—xn+iun+i+x20u1—xn+20un+1
(I —x)3 in general, and for the case of x=unity we have
1.4 +u2+ . . . +un=nul+n. 2—Dui+n.n—i.n—2o2u1+ .. . i. i.2.3
which will give the sum of the series very readily when un is a polynomial in n or a polynomial + a term of the form Ka".
4. Other types of series, whet' they can be summed to n terms at all, are summed by some special artifice. Summing the series to 3 or 4 terms may suggest the form of the sum to n terms which can then be established by induction. Or it may be possible to express
un in the form —wn, in which case the sum to n terms is w,Fi—w1.
Thus, if un=a(a+b)(a+2b) . . . (a+n7-ib)/c(c+b)(c+2b) . (c+n—Ib), the relation (c+nb)un+i=(a+nb)un can be thrown into the form (c+nb)un+1—(c+n—ib)un=(a—c+b)un, whence the sum can be found. Again, if un=tan nx tan (n+1)x, the summation can be effected by writing un in the form cot x (tan n + Ix — tan nx) — I. Or a series may be recognized as a coefficient in a product. Thus, if f(x)=fto+uix+u2x2+..., uo+ui+...+un is the coefficient of xn in f(x)/(r —x) ; in this way the sum of the first n coefficients in the expansion of (i —x)-k may be found. The sum of one series may be deduced from that of another by differentiation or integration. For further information the reader may consult G. Chrystal's Algebra (vol. ii.).
5. The sum of an infinite series may be deduced from the sum to n terms, when this is known, by increasing n indefinitely and finding the limit, if any, to which it tends, but a series may often be summed to infinity when it cannot be summed to n terms; the
sum of the infinite series I2 1 22+32+• • is 6, the sum to n terms cannot be found.
n terms of a series may be found approximately when it cannot be found exactly, the reader may consult G. Boole's Treatise on the Calculus of Finite Differences.
6. Let u1, u2, u3,...u,,, be a series of numbers real or complex, and let S. denote u1+u2+... +un. We thus form a sequence of numbers S1,S2, . . .Sn. This sequence may tend to a definite finite limit S as n increases indefinitely. In this case the series ul+u2+... +un is said to be convergent, and to converge to a sum S. If by taking n sufficiently large ISnI can be made to exceed any assignable quantity, however large, the series is said to be divergent. If the sequence Si, S2,... tends to finite but different limits according to the form of n the series is said to oscillate, and is also classed under the head of divergent series. The sum of n terms of the geometric series i+x+x2+.. .is (i—xn)/(i—x). If x is less than unity S. clearly tends to the limit i/(i —x), and the series is convergent and its sum is i/(i —x). If x is greater than unity S. clearly can be made gre ter than any assignable quantity by taking n large enough, and the %eries is divergent. The series i — i +1 — i + ... where Sn is unity or zero, according as n is odd or even, is an example of an oscillating series. The condition of convergency may also be presented under the following form. Let 5R,, denote Sn.f,—Sn: let e be any arbitrarily assigned positive quantity as small as we please; if we can find a number m such that for m=or>n, I,5Rnl Mn>S9; so that L. tends to the common limit of S, and So, which is the sum of the original series. If u1, u2, U3,... are all positive, and if after some fixed term, say the pig', u,, continually decreases and tends to the limit zero the series ui—u2+u3—u4+ . iS convergent. For IS,+2n—St,i lies between lu„+i—up+2 and lu,+i-up+2„I so that, when n is increased indefinitely, ISp+2ni remains finite; also ISP+2n+1—Se+2,I tends to zero, so that the series converges. If tin tends to a limit a, distinct from zero, then the series v1—v2+v3—.. ., where vn=un—a, converges and the series U1—u2+u3... oscillates. As examples we may take the series i — 2 + a — 4 + ... and 2 — z +i; — + .. , the first of these converges, the second oscillates.
9. The series ul+u3+u6+.. ., u2+u4+u4+... may each of them diverge, though the series ul—u2+u3—.. .converges. A series such that the series formed by taking all its terms positively is convergent is said to be absolutely convergent ; when this is not the case the series is said to be semi-convergent or conditionally convergent. A series of complex numbers in which un=pn+ign, where Ps and qn are real (i being Al -1), is said to be convergent when the series pi+p2+p3+..., gi+g2+q3+•.. are separately convergent, and if they converge to P and Q respectively the sum of the series is P+iQ. Such a series is said to be absolutely convergent when the series of moduli of un, i.e., E(pn2+gn2)l, is convergent; this is sufficient but not necessary for the separate convergence of the p and q series.
There is an important distinction between absolutely convergent and conditionally convergent series. In an absolutely convergent series the sum is the same whatever the order of the terms; this is not the case with a conditionally convergent series. The two series 1—1+i—1 +..., and 1+a—+i+-i . ., in which the terms are the same but in different orders, are convergent but not absolutely convergent. If we denote the sum of the first by S and the sum of the second by E it can be shown that E _ gS. G. F. B. Riemann and P. G. L. Dirichlet have shown that the terms of a semi-convergent series may be so arranged as to make the series converge to any assigned value or even to diverge.
lo. Tests for convergency of series of positive terms are obtained
by comparing the series with some series whose convergency or
divergency is readily established. If the series of positive terms
ui+u2-j-u3-I-..., vi+v2+v3+• • . are such that un/vn is always
finite, then they are convergent or divergent together; if
un+i/unvn+i/vn and Evn is divergent, then 2',un is divergent. By
For methods and transformations by means of which the sum to comparison with the ordinary geometric progression we obtain the
following tests. If nllu,. approaches a limit 1 as n is indefinitely increased, 3. us will converge if l is less than unity and will diverge if l is greater than unity (Cauchy's test) ; if u,.a.i/u,. approaches a limit l as n is indefinitely increased, Eus will converge if 1 is less than unity and diverge if 1 is greater than unity (D'Alembert's test). Nothing is settled when the limit l is unity, except in the case when l remains greater than unity as it approaches unity. The series then diverges. It may be remarked that if u„i i/u,. approaches a limit
and 'lu„ approaches a limit, the two limits are the same. The choice of the more useful test to apply to a particular series depends on its form.
In the case in which u,.+l/us approaches unity remaining constantly less than unity, J. L. Raabe and J. M. C. Duhamel have given the following further criterion. Write u„/u,.+i = 1+a,., where an is positive and approaches zero as n is indefinitely increased. If na„ approaches a limit 1, the series converges for 1> I and diverges for 1< I. For 1= i nothing is settled except for the case where 1 remains constantly_Iess than unity as it approaches it; in this case the series diverges.
If f(n) is positive and decreases as n increases, the series If(n) is convergent or divergent with the series Za'f(a") where a is any number >2 (Cauchy's condensation test). By means of this theorem we can show that the series whose general terms are
I I I I
na' n(ln)a' nln(12n)a' nlnl2n(13n)a'" '
where In denotes log n,12n denotes log•log n, Pn denotes log log log n, and so on, are convergent if a> I and divergent if a =or< 1.
By comparison with these series, a sequence of criteria, known as the logarithm criteria, has been established by De Morgan and J. L. Bertrand. A. De Morgan's form is as follows: writing us =1/0(n), put po=x*'(x)/d'(x), pi=(po-I)lx, ps=(pi-1)12x, p3=(p2-1)13x,.. . where l'x denotes log log log. . .x. If the limit, when x is infinite, of the first of the functions Po, 1?i, p2,..., whose limit is not unity, is greater than unity the series is convergent, if less than unity it is divergent.
In Bertrand's form we take the series of functions
I I I
1-/In, lnunn°n' lnnnlnmmis,.. .
If the limit, when n is infinite, of the first of these functions, whose limit is not unity, is greater than unity the series is convergent, if less than unity it is divergent. Other forms of these criteria may be found in Chrystal's Algebra, vol. ii.
Though, sufficient to test such series as occur in ordinary mathematics, it is possible to construct series for which they entirely fail. It follows that in a convergent series not only must we have Lt u,,=o but also Lt nun =o, Lt nlnu,,=o,&c. Abel has, however, shown that no function 0(n) can exist such that the series Eun is convergent or divergent as Lt ¢(n)u„ is or is not zero.
i i. Two or more absolutely convergent series may be added together, thus (ul+u2+• . . ) +(vl+v2+. . .)=(ui +vl) + (n2+v2)+
., that is, the resulting series is absolutely convergent and has for its sum the sum of the sums of the two series. Similarly two or more absolutely convergent series may be! multiplied together thus (nl+n2+U3+...) (vl+v2+v3+...) =uiv1+(niv2+n2v1)+(uiv3+
U2v2+uell) +.. •,
and the resulting series is absolutely convergent and its sum is the product of the sums of the two series. This was shown by Cauchy, who also showed that the series nu, where w„=uiv,.+u2vs-i+ .. +u,.vi, is not necessarily convergent when both series are semi-
convergent. A striking' instance is furnished by the series I-22+
J 3 4+. ' • which is convergent, while its- square i - 22+
( 33+2) - ... may be shown to be divergent. F. K. L. Mertens
has shown that a sufficient condition is that one of the two series should be absolutely convergent, and Abel has shown that if Ew„ converges at all, it converges to the product of 2u,. and Ev,.. But more properly the multiplication of two series gives rise to a double series of which the general term is umv,,.
12. Before considering a double series we may consider the case of a series extending backwards and forwards to infinity
.. u_m + . . . +14—2 +n—i +us +ui +u2 + ... +us+ ...
Such a series may be absolutely convergent and the sum is then
independent of the order of the terms and is equal to the sums of the
two series us-dui+u2+... and u-i+u-24-..., but, if not absolutely
convergent, the expression has no definite meaning until it is
explained in what manner the terms are intended to be grouped
together; for instance, the expression may be used to denote the
foregoing sum of two series, or to denote the series uo+(ui+u-I)+
(u2+u-2)+..., and the sum may have different values, or there
may be no sum, accordingly. Thus, if the series be ... -1-1+
o+i+z+..., with the former meaning the two series o+i+s+
. and -1-1- ... are each divergent, and there is no sum; but with the latter meaning the series is o+o+o+... which has a sum o. So, if the series be taken to denote the limit of (uo+ui+... +u„) -+-(u_i+u_ +.. . +u- ), where n and m are each of them ultimatelyinfinite, there may be a sum depending on the ratio n : m, which sum acquires a determinate value only when this ratio is given. In the case of the series given above, if this ratio is k, the sum of the series is log k.
13. In a singly infinite series we have a general term u,., where n is an integer positive in the case of an ordinary series, and positive or negative in the case of a back-and-forwards series. Similarly for a doubly infinite series we have a general term um,,. where m, n are integers which may be each of them positive, and the form of the series is then
Uo,o, Uo,l, Uo,2, .. .
Ul,o, Ul,l, Ui,z,...
or they may be each of them positive or negative. The latter is the more general supposition, and includes the former, since um,,. may =o, for m or n each or either of them negative. To attach a definite meaning to the notion of a sum, we may regard m, n as the rectangular coordinates of a point in a plane; if m and n are each positive we attend only to the positive quadrant of the plane, but otherwise to the whole plane. We may imagine a boundary depending on a parameter T, which for T infinite is at every point thereof at an infinite distance from the boundary; for instance, the boundary may be the circle x2+yi=T, or the four sides of a rectangle, x = =aT, y = T. Suppose the form is given and the value of T, and let the sum S,a,,, be understood to denote the sum of the terms um,,. within the boundary, then, if as T increases without limit, 5,.,,. continually approaches a determinate limit (dependent, it may be, on the form of the boundary) for such form of boundary the series is said to be convergent, and the sum of the doubly infinite series is the limit of Sm,,.. The condition of convergency may be otherwise stated; it must be possible to take T so large that the sum Rm,, for all terms um,,. which correspond to points outside the boundary shall be as small as we please.
14. It is easy to see that, if each of the terms um,,. is positive and the series is convergent for any particular form of boundary, it will be convergent for any other form of boundary, and the sum will be the same in each case. Suppose that in the first case the boundary is the curve fi(x, y) =T. Draw any other boundary f2(x, y) =T . Wholly within this we can draw a curve fl(x, y) =Ti of the first family, and wholly outside it we can draw a second curve of the first family, fi(x, y) =T2. The sum of all the points within f2(x, y) =T' lies between the sum of all the points within fi(x, y) =Ti and the sum of all the points within fi(x, y) =T2. It therefore tends to the common limit to which these two last sums tend. The sum is therefore independent of the form of the boundary. Such a series is said to be absolutely convergent, and similarly a doubly infinite series of positive and negative terms is absolutely convergent when the series formed by taking all its terms positively is convergent.
15. It is readily seen that when the series is not absolutely convergent the sum will depend on the form of the boundary. Consider the case in which m and n are always positive, and the boundary is the rectangle formed by x=m, y=n, and the axes. Let the sum within this rectangle be Sm,,.. This may have a limit when we first make n infinite and then m; it may have a limit when we first make m infinite and then n, but the limits are not necessarily the same; or there may be no limit in either of these cases but a limit depending on the ratio of m to n, that is to say, on the shape of the rectangle.
When the product of two series is arranged as a doubly infinite series, summing for the rectangular boundary x= aT, y =/3T we obtain the product of the sums of the series. When we arrange the double series in the form uivi+(uivz+uivo+• . . we are summing over the triangle bounded by the axes and the straight line x+y=T, and the results are not necessarily the same if the terms are not all pcsitive. For full particulars concerning multiple series the reader may consult E. Goursat, Colas d'analyse, vol. i.; G. Chrystal, Algebra, vol. ii.; or T. J. I'A. Bromwich, The Theory of Infinite Series.
16. In the series so far considered the terms are actual numbers, or, at least, if the terms are functions of a variable, we have considered the convergency only when that variable has an assigned
value. In the case, however, of a series ui(z)+u2(z)+..., where
ul(z), u2(z),... are single-valued continuous functions of the general complex variable z, if the series converges for any value of z, in general it converges for all values of z, whose representative points lie within a certain area called the " domain of convergence " and within this area defines a function which we may call S(z). It might be supposed that S(z) was necessarily a continuous function of z, but this is not the case. G. G. Stokes (1847) and P. L. Seidel (1848) independently discovered that in the neighbourhood of a point of discontinuity the convergence is infinitely slow and thence arises the notion of uniform and non-uniform convergence.
17. If for any value of z the series ui(z)+U2(z)+...converges it is possible to find an integer n such that I S(z) -S,.(z)I I.
A power series converges absolutely and uniformly at every point within its circle of convergence; it may be differentiated or integrated term by term; the function represented by a power series is continuous within its circle of convergence and, if the series is convergent on the circle of convergence, the continuity extends on to the circle of convergence. Two power series cannot be equal throughout any region in which both are convergent without being identical.
19. Series of the type ao+ai cos z+az cos 2z+ .. .
+bl sin z+bz sin 2z+ ..
where the coefficients ao, al, az, . . . bl, bz, . . . are independent of z, are called Fourier's series. They are of the greatest interest and importance both from the point of view of analysis and also because of their applications to physical problems. For the consideration of these series and the expansion of arbitrary functions in series of this type see FUNCTION and FOURIER'S SERIES. For the general problem of the development of functions in infinite series of various types
20. The modern theory of convergence dates from the publication in 1821 of Cauchy's Analyse algebrique. The great mathematicians of the 18th century used infinite series freely with very little regard to their convergence or divergence and with, occasionally, very extraordinary results. Series which are ultimately divergent may be used to calculate values of functions in special cases and to represent what are called " asymptotic expansions " of functions (see
21. The product of an infinite number of factors formed in succession according to any given law is called an infinite product. The infinite product Ho.= (I +u1) (i +uz) . . . (1 +u„) is said to be convergent when Ltn_,;TT, tends to a definite finite limit other than zero. If Lt IIn is zero or infinite or tending to different finite values according to the form of n the product is said to be divergent.
The condition for convergency may also be stated in the following form. (1) The value of IIn remains finite and different from zero however great n may become, and (2) Lt IL and Lt L+,. must be equal, when n is increased indefinitely, and r is any positive integer. Since in particular Lt II„ = Lt II,.+i, we must have Lt u,.+1 =o. Hence after some fixed term u1, uz, . . . or their moduli in the case of complex quantities, must diminish continually down to zero. Since we may remove any finite number of terms in which lu„1> 1 without affecting the convergence of the whole product, we may regard as the general type of a convergent product (I+ui)(I+uz) . . . (I +u,.) ... where lull, nil, . . . Iunl, ...are all less than unity and decrease continually to zero.
A convergent infinite product is said to be absolutely convergent where the order of its factors is immaterial. Where this is not the case it is said to be semi-convergent.
22. The necessary and sufficient condition that the product (i+ui)(i+uz) . . . should converge absolutely is that the series lull +17421+ . . . should be convergent. If u1, u2, . . . are all of the same sign, then, if the series u1+uz+ . . . is divergent, the product is infinite if ui,uz, . . . are all positive and zero if they are all negative.
If u1+uz+ . . . is a semi-convergent series the product converges, but not absolutely, or diverges to the value zero, according as the series u12+u22+ . . . is convergent or divergent. These results may671
be deduced by considering, instead of II,,, log H" which is the series log (I+ui)+log (I+uz)+ . . . (see G. Chrystal's Algebra, vol. ii., or E. T. Whittaker's Modern Analysis, chap. ii.); they may also be proved by means of elementary theorems on inequalities (see E. W. Hobson's Plane Trigonometry, chap. xvii.).
23. If u1, u2, ... are functions of a variable z, a convergent infinite product (I +u1) (I +uz) . . . defines a function of z. For such products there is a theory of uniform convergence analogous to that of infinite series. Is is not in general possible to represent a function as an infinite product; the question has been dealt with by Weierstrass (see his Abhandlungen aus der Functionlehre or A. R. Forsyth's Theory of Functions). One of the simplest cases of a function ex-pressed as an infinite product is that of sin z/z, which is the value of the absolutely convergent infinite product.
zz r zz / z2
I -7r- ) (I -227rz) ... (I -n7rz 1 . .
24. K. T. W. Weierstrass has shown that a semi-convergent or divergent infinite product may be made absolutely convergent by the association with each factor of a suitable exponential factor called
sometimes a " convergency factor." The product (1+1) (1+2—z7) (1+1) ... is divergent ; the product (1+9 e " (1+1 -) e 2"
is absolutely convergent. The product for sin z/z is semi-convergent when written in the form
(I 7r) (1 1 7r (I 27r) (I+ 7r) ...,
but absolutely convergent when written in the form
(I 7r) e" (I +7r) e " (1 2;) e2. (1+2Z7) a 2" ..
From this last form it can be shown that if
o(z)= (I—7r (I 27r) .. . (I—n7r) (I+7) (I+27) ... (I+m7r)
then the limit of rp(z) as m and n are both made infinite in any given ratio is
Another example of an absolutely convergent infinite product, whose convergency depends on the presence of an exponential
factor, is the product zII (1—z— ) ea ‘22' where f denotes 2mw1+
2nwz, col and wz being any two quantities having a complex ratio, and the product is taken over all positive and negative integer and zero values of m and n, except simultaneous zeros. This product is the expression in factors of Weierstrass's elliptic function 0(z).
A Course of Pure Mathematics. (A. E. J.)
End of Article: SERIES (a Latin word from serere, to join)