1911 Encyclopædia Britannica/Series

by Arthur Ernest Jolliffe

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 determined by its position, is said to form a series. Usually a series proceeds in one direction and the successive terms are denoted by $u_{1},u_{2},\ldots u_{n},\ldots$; we may, however, have a series proceeding in both directions, a back-and-forwards series, in which case the terms are denoted by

$\ldots u_{-m},\ldots u_{-2},u_{-1},u_{0},u_{1},u_{2},\ldots u_{n},\ldots$;

or its general term may depend on two integers positive or negative, and its general term may be denoted by $u_{m,n}$; 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.

Finite Series.

1. 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 which the general term is $2^{n}$; it may equally well be the series of which the general term is ${\frac {1}{6}}(n^{3}+5n+6)$; 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

$a_{p}u_{r}+a_{p-1}u_{r+1}+a_{p-2}u_{r+2}+\ldots +a_{1}u_{r+p-1}+a_{0}u_{r+p}=0$,

the series is said to be a recurring series with a scale of relation $a_{0}+a_{1}x+a_{2}x^{2}+\ldots \ +a_{p}x^{p}$. It is clear that we can regard the series $u_{0}+u_{1}x+u_{2}x^{2}\ldots \$ as the expansion in powers of $x$ of an expression of the form

$(b_{0}+b_{1}x+\ldots \ b_{p-1}x^{p-1})/(a_{0}+a_{1}x+\ldots \ a_{p}x^{p})$

,

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 $\Sigma \phi (n)\alpha ^{n}$, where $\phi (n)$ is a rational integral algebraic function of $n$, and $\alpha$ is independent of $n$. The series whose general term is of the form $K\alpha ^{n}+\phi (n)$, where $\phi (n)$ is a rational integral algebraic function of degree $r$, is a recurring series whose scale of relation is $(1-\alpha x)(1-x)^{r+1}$, but the general term of this series may be obtained by another method. Suppose we have a series $u_{0},u_{1},u_{2},\ldots \$ From this we can form a series $v_{0},v_{1},v_{2},\ldots \$ where $v_{n}=u_{n+1}-u_{n}$; from $v_{0},v_{1},v_{2},\ldots \$ we similarly form another series and so on; we write $v_{n}\equiv \Delta u_{n}$, and we suppose $E$ to be an operation such that $Eu_{n}\equiv u_{n+1}$ (the notation is that of the calculus of finite differences); the operations $E$ and $1+\Delta$ are equivalent and hence the operations $E^{n}$ and $(1+\Delta )^{n}$ are equivalent, so that we obtain $u_{n}=u_{0}+n\Delta u_{0}+{\frac {n\cdot n-1}{1\cdot 2}}\Delta ^{2}u_{0}\ldots \$ This is true whatever the form of $u_{n}$. When $u_{n}$ is of the form $K\alpha ^{n}+\phi (n)$, where $\phi (n)$ is of degree $r$, $\Delta ^{r+1}u_{0}$, $\Delta ^{r+2}u_{0}$, $\ldots \$ form a geometrical progression, of which the common difference is $\alpha -1$, or vanish if the term $K\alpha ^{n}$ is absent. In either case we readily obtain the expression for $u_{n}$.

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,\ldots \$ is $na+{\frac {1}{2}}n(n-1)b$; the sum of $n$ terms of the geometric progression $a,ab,ab^{2},\ldots \$ is $a(1-b^{n})/(1-b)$; yet we can find no simple expression to represent the sum of $n$ terms of the harmonic progression

$1+{\frac {1}{2}}+{\frac {1}{3}}+\ldots \ +{\frac {1}{n}}$.

3. The only type of series that can be summed to $n$ terms with complete generality is a recurring series. If we let $S_{n}=u_{0}+u_{1}x+\ldots \ +u_{n-1}x^{n-1}$ where $u_{0},\ldots \$ is a recurring series with a given scale of relation, for simplicity take it to be $1+px+gx^{2}$, we shall have

$S_{n}(1+px+gx^{2})=u_{0}+(ui_{1}+pu_{0})x+(pu_{n-1}r+qu_{n-2})x^{n}+qu_{n+1}x^{n+1}$.

If $x$ had a value that made $1+px+gx^{2}$ 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 unx" we have from the equivalence of the operations E and 1+∆,

urx+u2x2+ -i-ufrx

n xul - x'*+1u, tt1 +x2Au1 -x"+2Au, .+1

1 - x (1 - x)”

x3A2u1 xn+3A2u"+1

in general, and for the case of x=unity we have

u₁+u₂+ . . . + u_n＝nu₁+n.n/1.2∆u₁+n.n − 1.n − 2/1.2.3∆²u₁ + . . .,

which will give the sum of the series very readily when u_n is a polynomial in n or a polynomial + a term of the form Kαⁿ.

4. Other types of series, when 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 u, . in the form w, .+1-wn, in which case the sum to n terms is w»+1-wr. Thus, if u, .=a(a-}-b)(a+2b) (a-{-n-Ib)/c(c+b)(c+2b) (c-i-n-lb), the relation (c+nb)u, ,,1=(a+nb)u, . can be thrown into the form (c+nb)u, .+1-(c+n-1b)u, .=(a-c-4-b)u, ., whence the sum can be found. Again, if u, ,=tan nx tan (n+r)x, the summation can be effected by writing un in the form cot x (tan n+ rx -tan nx) - I-Or a series may be recognized as a coefficient in a product. Thus, if f(>C)EHo-l-141964-142062-l- ., ao-I-ul-I-.. +uf» is the coefficient of x" in f(x)/(I -x); in this way the sum of the first n coefficients in the expansion of (1−x)^−k may be found. The sum of one series may be deduced from that of another by differentiation or integration. Forlfumher 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 rt 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 %-I-%+§ ;-I- .is § , the sum to n terms cannot be found.

For methods and transformations by means of which the sum to 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.

Infinite Series.

6. Let ur, 'u¢, u3, u, , be a series of numbers real or complex, and let S.. denote u1+u2+. +u, .. We thus form a. sequence of numbers Sr, S2, S". This sequence may tend to a definite finite limit S as ri increases indefinitely. In this case the series u1+u2+. -l-un is said to be convergent, and to converge to a sum S. If by taking n sufficiently large |S, ,| can be made to exceed any assignable quantity, however large, the series is said to be divergent. If the sequence S1, 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-l-x-1-x2+ is (1-x )/(1-x). If x is less than unity S, , clearly tends to the limit I /(1 -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 thexeries is divergent. The series I -I -I-1 - I -{- ., where S, . is unity or zero, according as n is odd or even, is an example of an oscillating series. The condition of convergence may also be presented under the following form. Let, ,Rf denote S, ,, H, -Sn: let e be any arbitrarily assigned 'positive quantity as small as we please; if we can find a number rn such that for m=or>n, k, R, ,|<e for all values I, 2, . of p, then the series-converges. T e least value of the number in corresponding to a given value of e, if it can be found, may be regarded as a measure of rapidity of the convergence of the series; it may happen that when un involves a variable x, rn increases indefinitely as x approaches some value; in this case the convergence of the series is said to be infinitely slow for this value of x.

7. An infinite series may contain both positive and negative terms. The terms may be positive and negative alternately or they may occur in groups which without altering the order of the terms of the series may each be collected into a single term; thus all series may be regarded as belonging to one of two types, ul-i-u2+u3+ in which the terms are all positive, or u1-u2-|-u3- in which the terms are alternately positive and negative. 8. It is clear that if a series is convergent un must tend to the limit zero as n is increased indefinitely. This condition though necessary is by no means sufficient. If all the termsof a convergent series are positive a series obtained by writing its terms in any other order is convergent and converges to the same sum. For if Sn denotes the sum of ri terms of the first series and En denotes the sum of n terms of the new series, then, when n is any large number, we can choose numbers p and q such that Sq>2, .>S, ,; so that 2, . tends to the common limit of Sp and Sq, which is the sum of the original series. If ul, ng, ua, are all positive, and if after some fixed term, say the pf", ii.. continually decreases and tends to the limit zero the series ul-ug-1-us-u4+ is convergent. For Sp, ”-SPI lies between [u, ,+1-u, ,+2 and |up+1-u, ,+2, .| so that, W en n IS increased indefinitely, , ,+2"| remains finite; also ]S, ,+2, .+1-S, ,+2, .| tends to zero, so that the series converges. If un tends to a limit a, distinct from zero, then the series v1-v2+v3- , where v, ,=u, ,-a, converges and the series ul-itz-l-ua. oscillates As examples We may take the series I-%~}-15-i+ and 2-g-|-§ -§ + . ; the first of these converges, the second oscillates 9. The series u1+u3+ii5+ ., u2+u4+u6-l- may each of them diverge, though the series 'u1-u2-{-u3- converges. A series such that the series formed by taking all its terms positively is convergent is saidto 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 u, ,=p, +ig, ., where pn and qu are real (i being / -I), is said to be convergent when the series ppl-pg-i-p3+ , qi-i-Q2-l-qa+. 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({>, ,2+g, ,2)é, 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 I -%-i-é-i-I- , and I+§ *%+§ ~+§ ~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 2 =§ S 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. >

IO. Tests for convergence of series of positive terms are obtained by comparing the series with some series whose convergence or divergence is readily established. If the series of positive terms ul-l-u2+u3+. ., vi--v2-l-va-- are such that un/v., is always finite, then they are convergent or divergent together; if u1..t1/un<v¢.+1/vu and Ev" is convergent, then Eu, ” is convergent; if u, .+1/u, .>v, +1/v, , and Ev" is divergent, then Eu" is divergent. By comparison with the ordinary geometric progression we obtain the following tests. If 'X/u, . approaches a limit l as n is indefinitely increased, Eu.. will converge if l is less than unity and will diverge if is greater than unity (Cauchy's test); if u, ,+1/u, . approaches a limit l as n is indefinitely increased, Eu.. will converge if l is less than unity and diverge if l is greater than unity (D'Alembert's test). Nothing is settled when the limit l is unity, except in the case when remains greater than unity as it approaches unity. The series then diverges. It may be remarked that if u, ,, ,1/u, , approaches a limit and 'i/un 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, .+1/un approaches unity remaining constantly less than unity, ] L. Raabe and J. M. C. Duhamel have given the following further criterion. Write u, ,/u, ., , 1=I+a, ., where af. IS positive and approaches zero as n is indefinitely increased. If nan approaches a limit l, the series converges for l>I and diverges for l< 1. For l= 1 nothing is settled except for the case where l remains constantly less than unity as it approaches it; in this case the series diverges /

If f (n) is positive and decreases as n increases, the series 2f(u) is convergent or divergent with the series 2a"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(1rL)¢' nln(l2n)f1' nln12nll“n)¢'° ' "

where in denotes log n, Pu denotes log-log n, lin denotes log log log n, and so on, are convergent if a> 1 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 ]. L. Bertrand. A. De Morgan's form is as follows: writing u, ,=1/4>(n), Put Po=x¢>'(x)/¢>(x), pl =(P0'°1)lx1P2=(P1~'{)l2x, P3=(P2"'1)l3xi- -where l'x denotes log log log. . x. If the limit, when x is infinite, of the first of the functions po, pi, pa, . ., 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 L L 2 .L 3

lun/ln, Inu”/l rl, lmtnm/l n,

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 oniy must we have Lt u, , =0 but also Lt nu" =0, Lt nlnu" =O, &c. Abel has, however, shown that no function ¢(n) can exist such that the series Eu" is convergent or divergent as Lt q'>(n)u, , is or is not zero.

11. Two or more absolutely convergent series may be added together, thus (ul-l-ug-i-. .)+(v1 +112-l- .) =(u1+v,)-|-(ua-|-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 to ether thus (u1"l'u2'i'u3'l'- - -) ('”1~l'Uz'l"Us-i"- - -) =“1'U1+(1¢17J2+1la7J1)'l“§ "1vs+ ugifz-|-url/1)-I- .,

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 2w, ., where 'w, .=u1'v, ,+u2v, ,, 1+ +u, .v1, is not necessarily convergent when both series are semi convergent. A striking' instance is furnishedgby the series I -11%-P é-$4- which is convergent, while its square I -é-P 2 1

(QT)-l-5) - . may be shown to be divergent, F. K. L. Mertens has shown that a sufiicientcondition is that one of the two series should be absolutely convergent, and Abel has shown that if Ewa converges at all, it converges to the product of Eu" and Ev". But more properly the multiplication of two series gives rise to a double series of which the general term is u, ,, 'v, ,. 12. Before considering a double series we may consider the case of a series extending backwards and forwards to infinity u, ,.+. . +u 2+u 1+m>+u1+wi+- - . -I-uf.+. .

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 ug-l-ui-l-ur-l- and u-1+u»-yi- . ., 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-l-(u1+u-|)+ (url-u-2 + , and the sum may have different values, or there may be no sum, accordingly. Thus, if the series be . -é-}+ 0-l-}-+5-l- ., with the former meaning the two series 0+}-l-§ -|and -1-é- are each divergent, and there is no sum; but with the latter meaning the series is o+0-l-0+- which has a Sum 0. So, if the series be taken to denote the limit of (+ +- - - + 1. -l(u..|+u

+. -i-u., ,,), where n and rn are each of tliiem ultimiatiely infinite, there may be a sum depending on the ratio n sm, 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 uf., where n IS an' integer posltive 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 u, ,, ,, , where rn, n are integers which may be each of them positive, and the form of the series is then

140,01 Hou, u0»21~ -uno,

um, 'l41y2»~ -or

they may be each of them positive or negative. The latter is the more general supposition, and includes the former, since u, ,, ,, , 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 rn, n as the rectangular coordinates of a point in a plane; if m and rr 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 +3/2 =T, or the four sides of a rectangle, x = iaT, y = ==/ST. Suppose the form is given and the value of T, and let the sum Sm, " be understood to denote the sum of the terms u, ,, ,, , within the boundary, then, if as T increases without limit, Sm, " continually a proaches 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 convergence may be otherwise stated; it must be possible to take T so large that the sum R, ,, ,, . for all terms u, ,, ,, , 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 u, ,, ,, . 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. Sup ose that in the first case the boundary is the curve f1(x, y) =T. lgraw any other boundary f2(x, y) =T Wholly within this we can draw a curve f, (x, y)=T1 of the first family, and wholly outside it we can draw a second curve of the first family, f, (x, y) =T2. The sum of all the points within f2(x, y) =T lies between the sum of all the points within f1(x, y) =-T1 and the sum of all the points within f1(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 Smut. This may have a limit when we first make n infinite and then rn; 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 rn 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 ==,8T we obtain the product of the sums of the series. When we arrange the double series in the form um-l~(u1v2+u»2z11)+. 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 positive. For full particulars concerning multiple series the reader may consult E. Goursat, Cours 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 convergence only when that variable has an assigned value. In the case, however, of -a series u, (z) -I-u2(z)+ ., where u, (z), u2(z), are sin le-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-umforrn convergence. g

17. If for any value of z the series u1(z) +u2(z)+ .converges it is possible to find an integer n such that |S(z)-S, .(z)<e, |S(z)-S, , t1(z) |< e, , where e is any arbitrarily assigned positive quantity, however small, For a given s the least value of rl will vary throughout any region from point to point of that region. It may, however, be possible to find an integer 1' which is a superior limit to all the values of rr. in that region, and we thus have, throughout this region, I S(z) -S1/(z)1 < e, ]'S (z) -Sv+, (z) |< e. .where z is any point in the region and v is a finite integer depending only on e and not on z. The series is then said to converge uniformly throughout this region.

If, as z approaches the value z₁, n increases as |z−z₁| diminishes and becomes indefinitely great as |z−z₁| becomes indefinitely small the series is said to be non-uniformly convergent at the point z₁.

A function represented by a series is continuous throughout any region in which the series is uniformly convergent; there cannot be discontinuity with uniform convergence; on the other hand there may be continuity and non-uniform convergence. If u₁(z) +u₂(z) +... is uniformly convergent we shall have ∫S(z)dz =∫u₁(z)dz+ ∫u2(z)dz+... along any path in the region of uniform convergence;'and we shall also have d/dzS(z)=d/dzu₁(z)+(-£1-u2(z)+ if the series t-%u1(z) -1-(-%u2(z) + . . . is uniformly convergent.

Uniform convergence is essentially different from absolute convergence; neither implies the other (see Function).

18. A series of the form a₀+a₁z+a₂z₂+ . . ., in which a₀, a₁,a₂. . . . are independent of 2, is called a power series.

In the case of a power series there is a quantity R such that the series converges if |z|<R, and diverges if |z| >R. A circle described with the origin as centre and radius R is called the circle of convergence# A power series may or may not converge on the circle of convergence. The circle of convergence may be of infinite radius as in the case of the series for s1nz, v1z. z-57+ z⁵/5! . . . In this case the series converges over the whole of the z plane. Or its radius may be zero as in the case of the series 1+1! z+2! z²+ . . ., which converges nowhere except at the origin. The radius R may be found usually, but not always, from the consideration that a series converges absolutely if |u, tt1/u.|<1, and diverges if |u_n+1/u_n| > 1.

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 a₀+a₁ cos z+a₂ cos 2z+ . . . +b₁ sin z+b₂ sin 2z + . . .,

where the coefficients a₀, a₁, a₂, . . . b₁, b₂, . . . 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 see Function.

20. The modern theory of convergence dates from the publication in 1821 of Cauchy's Analyse algébrique. 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 Function).

Infinite Products.

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 Π_n≡ (1 +u₁)(1+u₂) . . . (1 + u_n) is said to be convergent when Lt, ., ,IL. tends to a definite finite limit other than zero. If Lt IL. 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 convergence may also be stated in the following form. (1) The value of Π_n remains finite and different from zero however great n may become, and (2) Lt Π_n and Lt Π_n+r, must be equal, when n is increased indefinitely, and r is any positive integer. Since in particular Lt IL.=Lt IL, H, we must have Lt u, tt1=o. Hence after some fixed term ui, ug, 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 [u, .| > 1 without affecting the convergence of the whole product, we may regard as the general type of a convergent product (I+u1)(I -l-uz) . (1+u, .) where lu1|, Iuzl, |u, .|, 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 (1 +u1)(1+u2) should converge absolutely is that the series u1|+|u2-l- should be convergent. If ul, ug, are all of the same sign, then, if the series url-u2+ . is divergent, the product is infinite if u1, u2, .are all positive and zero if they are all negative. If ur-i-144+ is a semi-convergent series the product converges, but not absolutely, or diverges to the value zero, according as the series ui'-l-u2'+ is convergent or divergent. These results may be deduced by considering, instead of Hn, log IL. which is the series log (1 -l-ui) -I-log (I +u2)-l- . . (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 ui, u¢, are functions of a variable z, a convergent infinite product (I -l-u1)(I -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 expressed as an infinite product is that of sin z/z, which is the value of the absolutely convergent infinite product. Z2 Z2 Z2

(I-E) (I-5573> . . . (If-W .

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 “convergence factor.” The product <1 (I -Fi) -Z

(T +5;> is divergent; the product <1 -|-ij) 6 Tr (I-hi) e 2" is absolutely convergent. The product for sin z/z is semi-convergent when written in the form

z z z z

(1-r) (In) (1-sl (Im)

but absolutely convergent when written in the form Z E .l

(1-3) @"(1+§)e "(1-%) (1+i%.)e "

From this last form it can be shown that if Z

= E L l E L .

φ(z) <1 T) (I M) (I mr) <1-l-W) (1-l-21) . <1-l-mf),

then the limit of 4>(z) as m and rt are both made infinite in any given ratio is

(m) 5 sin z

1 'IT lil

Another example of an absolutely convergent infinite product, whose convergence depends on the presence of an exponential factor, is the product 2Π (1−z/Ω)^{z/eΩ+1/2z2/Ω2} where Ω denotes 2mω₁+ 2nω₂, ω₁au and ω₂ 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 σ(z).

Authorities.—G. Chrystal, Algebra, vol. ii. (1900); E. Goursat, Cours d’analyse (translated by E. R. Hedrick), vol. i. (1902); J. Harkness and F. Morley, A Treatise on the Theory of Functions (1893) and Introduction to the Theory of Analytic Functions (1899); E. W. Hobson, Plane Trigonometry (1891), and Theory of Functions of a Real Variable: H. S. Carslaw, Fourier’s Series; E. Whittaker, Modern Analysis (1902); J. Tannery, Introduction à la théorie des fonctions d'une variable; C. Jordan, Cours d'analyse de l’École Polytechnique (2nd ed., 1896); E. Césaro, Corso di analisi algebraica (1894); O. Stolz, Allgemeine Arithmetik (1886); O. Biermann, Elemente der höheren Mathematik (1895); W. F. Osgood, Introduction to Infinite Series; T. J. I’A. Bromwich, Theory of Infinite Series (1908). Also the article by A. Pringsheim, “Irrationalzahlen und Konvergenz unendlichen Prozesse” in the Encyclopädie der mathematischen Wissenschaften 1, a. 3 (Leipzig). For the history of the subject see R. Reiff, Geschichte der unendlichen Reihen; G. H. Hardy, A Course of Pure Mathematics.  (A. E. J.)