terms convergent positive limit infinite absolutely divergent negative term function
SERIES. A series is a set of terms considered as arranged in order. Usually the terms are or represent numerical magnitudes, and we are concerned with the sum of the series. The number of terms may be limited or without limit ; and we have thus the two theories, finite series and infinite series. The notions of convergency and divergency present themselves only in the latter theory.
will be 1, 2, 4, 8, 15, 26, .. The series may contain negative terms, and in forming the sum each term is of course to be taken with the proper sign.
will, it may be, suggest (but it cannot do more than suggest) the expression for the sum of n terms of the series. For instance, for the series of odd numbers 1 + 3 + 5 +7 +... , we have 1 = 1, 1 + 3=4, 1 + 3 + 5 =9, &e. These results at once suggest the law, 1 + 3 + 5 . + (2n - 1)= n2, which is in fact the true expression for the sum of a terms of the series ; and this general expression, once obtained, can afterwards be verified.
Simple cases are the three which follow.
The arithmetic series, a+(a+b)+(a+2b)..+(a+n-1)b; writing here the terms in the reverse order, it at once appears that twice the sum is = 2a + n - lb taken n times : that is, the sum =na + ln(n - 1)b. In particular we have an expression for the sum of the natural numbers 1+2+3...+n=--a(a+1), 2 and an expression for the sum of the odd numbers 1+3+5 ..+(2n - 1)=n2.
The geometric series, a+ ar+ar2 ..+arn-1 ; = a1 - • in particular the sum of the series 1+r+r2..+r"1=1 - rn 1-r • But the harmonic series, or say + + . . . + -Th, does not admit of summation ; there is no algebraical function of n which is equal to the sum of the series.
uo+u,...+11/4=-0 +1+3 ..+-2v.(n+1); here, observing that n(n+ 1)(n+ 2)- (n -1)n(n + 1)=n(n+1)(n + 2 - n - 1),= 32(91 + 1), _1 we have r„4.1- verb + 1)(n + 2) ; 1 and hence 1+3+6 + 2-n(n+i).-6n(it +1)(Pt + 2), as may be at once verified for any particular value of n. Similarly, when the general term is a factorial of the order r, we have r +1 + 91(n + 1) . . (II + r-1)=. n(n + 1) . . (91 +r) 1.2 r 1.2 .. (r+1) If the general term un be any rational and intedral function of n, we have at 91(n-1) *a -1) .. (a - p +1) p (a +1)a +-an= (n + 1)u, + 2 Au, ..
(n +1)n(n - 1) .. (n - p +1) 2) . 1.2.3..0)+1) which is a function of the degree p +1.
Thus for the before-mentioned series 1 + 2 + 4 + S + if it be assumed that the general term un is a cubic function of n, and writing down the given terms and forming the differences, 1, 2, 4, 8 ; 1, 2, 4 ; 1, 2 ; 1, we have -6(77 5n, A- 6), as above ; and the sum no +u, +un (91+1)n (n + 1)1p(n -1) (n+ 1)n(n - 1)(n - 2) =n+1+- 24(71,4+ 2n3+11n3+34n+24).
As particular cases we have expressions for the sums of the powers of the natural numbers - 12+22... +n2=-6n(n+1)(2n+ 1); 13+23.. +n3=-n 3(n+1)9 4 (observe that this = (1 + 2 + a)2); and so on.
We may, from the expression for the sum of the geometric series, obtain by differentiation other results thus l+r+r2...+rn--=1-r gives d 1 - rn 1 - 'arm + (91 - 1)rn dr 1-r (1 - r)2 and we might in this way find the sum /20+ /sir. +unr", where an is any rational and integral function of a.
- 1)x2+ . .) (14-itx-Fn(121)xt ..) 1.
=1+ (m +n)x+ (7n + n)(m IL - 1) X1+ . .
really means the series of identities (m+n)=m+n (m+n)(m+n-1)_. 9n(ni - I) n 91.(91 - I) obtained by multiplying together the two series of the left-hand side. Again, in the method of generating functions we are concerned with an equation 4,(e)= Ao+ Alt... + Anti' + . . , where the function CO is used only to express the law of formation of the successive coefficients.
It is an obvious remark that, although according to the original definition of a series the terms are considered as arranged in a determinate order, yet in a finite series (whether the number of terms be definite or indefinite) the sum is independent of the order of arrangement.
S. We consider an infinite series u0+ +7e2+ . . . of terms proceeding according to a given law, that is, the general term u„ is given as a function of it. To fix the ideas the terms may be taken to be 'positive numerical magnitudes, or say numbers continually diminishing to zero; that is, u.> u„,fi, and u. is, moreover, such a function of it that by taking is sufficiently large u,, can be made as small as we please.
Forming the successive sums S0 = u„ S1=210+ u„ S2 = uo+ ul+ u„ .. these sums So, Sp S2 . . : will be a series of continually increasing terms, and if they increase up to a determinate finite limit S (that is, if there exists a determinate numerical magnitude S such that by taking is sufficiently large we can make S - S,, as small as we please) S is said to be the sum of the infinite series. To show that we can actually have an infinite series with a given sum S, take uo any number less than S, then S - uo is positive, and taking xi any numerical magnitude less than S - uo, then S - u, - sti is positive. And going on continually in this manner we obtain a series uo+ul +u2+ . . . such that for any value of is however large S- uo - ... - un is positive ; and if as n increases this difference diminishes to zero, we have uo + u, + u2+ , - an infinite series having S for its sum. Thus, if S = 2, and we take u, <2, say it, = 1; u,< 2 - 1, say u1=1; u2<2 • - 1 - sayito= i; and so on, we have 1 + 2- + 4- -I- =2; 1 r r<1, has the finite sum 1 . This in fact follows from the expression 1 +r + r2... = r for the sum of the finite series ; taking r< 1, then as is increases 74' decreases to zero, and the sum becomes more and more 1 An infinite series of positive numbers can, it is clear, have a sum only if the terms continually diminish to zero ; but it is not conversely true that, if this condition be satisfied, there will be a sum. For instance, in the case of the harmonic series 1 +-23 + - + ... it can be shown that by tak- ing a sufficient number of terms the sum of the finite series may be made as large as we please. For, writing the series _L 7 \ in the form 1 1.+(1+ k5± 8) , the number of terms in the brackets being doubled at each successive step, it is clear that the sum of the terms in any bracket is always ›••1 ; hence by sufficiently increasing the number of brackets the sum may be made as large as we please. In the foregoing series, by grouping the terms \ in a different manner 1+-rk2 3)_L k4 ' 5÷ 6_L7 • • , the sum of the terms in any bracket is always <1 ; we thus arrive at the result that (n = 3 at least) the sum of 24 terms of the series is > 1 +n and <n.
2 An infinite series may contain negative terms ; suppose in the first instance that the terms are alternately positive and negative. Here the absolute magnitudes of the terms must decrease down to zero, but this is a suffi dent condition in order that the series may have a sum. The case in question is that of a series vo - yr + v2- , where vo, v1, v2, . are all positive and decrease down to zero. Here, forming the successive sums So = vo, S1= vo - v1, vo- v1+ v2, . . So, S, . . . are all positive, and we have So > Sl, Sl <S20 82 > 82, .. and S.44 - S. tends continually to zero. , Hence the sums So, S„ S20 .. tend continually to a positive limit S in such wise that So, S2, S4, . are each of them greater and Sp S„ S„ . are each of them less than S; and we thus have S as the sum - - + .. will serve as an example. The case just considered includes the apparently more general one where the series consists of alternate groups of positive and negative terms respectively ; the terms of the same group may be united into a single term v. and the original series will have a sum only if the resulting series vo + v2 ... has a sum, that is, if the positive partial sums vo, v1, v2, .. decrease down to zero.
The terms at the beginning of a series may be irregular as regards their signs ; but, when this is so, all the terms in question (assumed to be finite in number) may be united into a singl,, term, which is of course finite, and instead of the original series only the remaining terms of the series need be considered. Every infinite series whatever is thus substantially included under the two forms, - terms all positive and terms alternately positive and negative.
In brief, the sum (if any) of the infinite series + + u2 + .. is the finite limit, (if any) of the successive sums uo, uou0+ M + u2, . ; if there is no such limit, then there is no sum. Observe that the assumed order uo, ul, u2... of the terms is part of and essential to the definition ; the terms in any other order may have a different sum, or may have no sum. A series having a sum is said to be " convergent " ; a series which has no sum is "divergent."
If a series of positive terms be convergent, the terms cannot, it is clear, continually increase, nor can they tend to a fixed limit : the series 1 + 1 + 1 + .. is divergent. For the convergency of the series it is necessary (but, as has been shown, not sufficient) that the terms shall decrease to zero. So, if a series with alternately positive and negative terms be convergent, the absolute magnitudes cannot, it is clear, continually increase. In reference to such a series Abel remarks, " Peut-on imaginer rien de plus horrible que de debiter 0 = In - 24 + 3m - 4n +, &c., ou is est un nombre entier positif?" Neither is it allowable that the absolute magnitudes shall tend to a fixed limit. The so-called "neutral" series 1 -1 + 1 - 1 .. is divergent : the successive sums do not tend to a determinate limit, but are alternately + 1 and 0 ; it is necessary (and also sufficient) that the absolute magnitudes shall decrease to zero.
In the so-called semi-convergent series we have an equation of the form S= U,- Ul + U2 - ... , where the positive values Uo, U1, U„ . . . decrease to a minimum value, suppose Up, and afterwards increase ; the series is divergent and has no sum, and thus S is not the sum of the series. S is only a number or function calculable approximately by means of the series regarded as a finite series terminating with the term f Up. The successive sums U0, U1, Uo - U1 + U2, .. up to that containing ± Up, give alternately superior and inferior limits of the number or function S.
definitely small ; by taking m sufficiently large the sum um+1+21m.).2 . . . . + vm+,. (where r is any number however large) can be made as small as we please ; or, as this may also be stated, the sum of the infinite series v„,+1+9.4.2+ . . . can be made as small as we please. If the terms are all positive (but not otherwise), we may take, instead of the entire series u„,4.1+ Um+2 ., any set of terms (not of necessity consecutive terms) subsequent to um; that is, for a convergent series of positive terms the sum of any set of terms subsequent to um can, by taking nt sufficiently large, be made as small as we please.
It follows that in a convergent series of positive terms the terms may be grouped together in any manner so as to form a finite number of partial series which will be each of them convergent, and such that the sum of their sums will be the sum of the given series. For instance, if the given series be u„ + u, +u2+..., then the two series 210+212+ v4+. . . and ?Li+ Rt3+ . . will each be convergent and the sum of their sums will be the sum of the original series.
Obviously the conclusion does not bold good in general for series of positive and negative terms : for instance, the series 1 - -1+-1 - -1+ .. is convergent, but the two series 1 + 3 - + - 1 1 • and - - - .. are each divergent, and thus without a sum. In order that the conclusion may be applicable to a series of positive and negative terms the series must be " absolutely convergent," that is, it must be convergent when all the terms are made positive. This implies that the positive terms taken by themselves are a convergent series, and also that the negative terms taken by themselves are a convergent series. It is hardly necessary to remark that a convergent series of positive terms is absolutely convergent. The question of the convergency or divergency of a series of positive and negative terms is of less importance than the question whether it is or is not absolutely convergent. But in this latter question we regard the terms as all positive, and the question in effect relates to series containing positive terms only.
Consider, then, a series of positive terms u0-1-9/1 + v2 + ..; if they are increasing - that is, if in the limit 21/44.1/u. be greater than 1 - the series is divergent, but if less than 1 the series is convergent. This may be called a first criterion; but there is the doubtful case where the limit =1. A second criterion was given by Cauchy and Raabe ; but there is here again a doubtful case when the limit considered =1. A succession 'of criteria was established by De Morgan, which it seems proper to give in the original form; but the equivalent criteria established by Bertrand are somewhat more convenient. In what follows lx is for shortness written to denote the logarithm of x, no matter to what base. De Morgan's form is as follows : - Writing u„=•;f•-(), put po = x • if for x= co the limit a, of Po be greater than 1 the series is convergent, but if less than 1 it is divergent. If the limit a0=1, seek for the limit of p1, = (1)2- 1)1z; if this limit al be greater than 1 the series is convergent, but if less than 1 it is divergent. If the limit al =1, seek for the limit p2, = (p1-1)11x ; if this limit a2 be greater than 1 the series is convergent, but if less than 1 it is divergent. And so on indefinitely.
Bertrand's form is : - If, in the limit for n = co, /-1lin, be negative or less than 1 the series is divergent, but if greater than 1 it is convergent. If it =1, then if / --1int be negative or less than 1 the series is divergent, but if greater than 1 it is convergent. And so on indefinitely.
The last-mentioned criteria follow at once from the theorem that the several series having the general terms .. respectively are each na 91(1a)a' nlnyln)a' nInlin(111-n)a of them convergent if a be greater than 1, but divergent if a be negative or less than 1 or = 1. In the simplest case, series with the general term the theorem may be proved nearly in the manner in which it is shown above (cf. § 9) that the harmonic series is divergent.
Two or more absolutely convergent series may be added together, ( u, + u1 +942 . + fv, + v1 +9,2 .. = (u, + v0) + (u1 + v1) ; that is, the resulting series is absolutely convergent and has for its sum the sum of the two sums. And similarly two or more absolutely convergent series may be multiplied together (Ito+ ul +u, . .) x (v0 + v1 + v2. .) =u0v9+(u0ni +92,9/0+ (u0v2+uivI+9/2v0)+ ; that is, the resulting series is absolutely convergent and has for its sum the product of the two sums. But more properly the multiplication gives rise to a doubly infinite seriesItovo, uovi, 4101)2• • • nom ufv„ 211v2 - which is a kind of series which will be presently considered.
1 Thus, if the series be . . - -2 - -1+ 0 +1 +2 + . . , in the former meaning the two series 0 +1+ .. and - - 7,1 - . .
are each divergent, and there is not any sum. But in the latter meaning the series is 0 + 0 + 0 + , which has a sum =0. So, if the series be taken to denote the limit of (u, + + u2 . . + um) + (u_1+ u_ 2 . .+u_,,,'),where an, ne are each of them ultimately infinite, there may be a sum depending on the ratio m : nt', which sum consequently acquires a determinate value only when this ratio is given.
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-andforwards series. Similarly for a doubly infinite series we have a general term um,„, where 9n, n are integers which may be each of them positive, and the form of the series is then , Ito? 1102 • • ' 113,0 > 111,1 iti,s or they may be each of them positive or negative. The latter is the more general supposition, and includes the former, since um,. may =0 for m or n each or either of them negative. To put a definite meaning on the notion of a sum, we may regard m, n as the rectangular coordinates of a point in a plane; that is, if 7/2, n are each of them positive we attend only to the positive quadrant of the plane, but otherwise to the whole plane ; and we have thus a doubly infinite system or lattice-work of points. We may imagine a boundary depending on a parameter T which for T=oo is at every point thereof at an infinite distance from the origin ; for instance, the boundary may be the circle x2 + y2 = T, or the four sides of a rectangle, x= y= 11371. Suppose the form is given and the value of T, and let the sum fu„,.,„ be understood to denote the sum of those terms which correspond to points within the boundary, then, if as T increases without limit the sum in question 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 aforesaid limit of the sum The condition of convergency may be otherwise stated : it must be possible to take T so large that the sum Ezt. 7, for all terms which correspond to points outside the boundary shall be as small as we please.
It is easy to see that, if the terms u„„ „ be all of them positive, and the series be 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. Thus, let the boundary be in the first instance the circle x2 + y2 = T; by taking T sufficiently large the sum 1'24;7, for points outside the circle may be made as small as we please. Consider any other form of boundary - for instance, an ellipse of given excentricity, - and let such an ellipse be drawn including within it the circle x2 + y2 = T. Then the sum for terms u„„,„ corresponding to points outside the ellipse will be smaller than the sum for points outside the circle, and the difference of the two sums - that is, the sum for points outside the circle and inside the ellipse - will also be less than that for points outside the circle, and can thus be made as small as we please. Hence finally the sum Iu,n n, whether restricted to terms 2‘,.,7,77 corresponding to points inside the circle or to terms corresponding to points inside the ellipse, will have the same value, or the sum of the series is independent of the form of the boundary. Such a series, viz., a doubly infinite convergent series of positive terms, is said to be absolutely convergent ; and similarly a doubly infinite series of positive and negative terms which is convergent when the terms are all taken as positive is absolutely convergent.
We have in the preceding theory the foundation of the theorem (§ 17) as to the product of two absolutely convergent series. The product is in the first instance expressed as a doubly infinite series ; and, if we sum this for the boundary x+ y= T, this is in effect a summation of the series uov, + (u0v, +u,v,)+ .. , which is the product of the two series. It may be further remarked that, starting with the doubly infinite series and summing for the rectangular boundary x = aT, y =13T, we obtain the sum as the product of the sums of the two single series. For series not absolutely convergent the theorem is not true. A striking instance is given by Cauchy : the series J4 • • -N/3 able sum, but it can be shown without difficulty that its square, viz., the series 1 - -22 + (3 -1 2) - . . , N/ is divergent.
The case where the terms of a series are imaginary comes under that where they are real. Suppose the general term is pn+ gni, then the series will have a sum, or will be convergent, if and only if the series having for its general term pi, and the series having. for its general term qn be each convergent ; then the sum = sum of first series + i into sum of second series. The notion of absolute conver gence will of course apply to each of the series separately ; further, if the series having for its general term the modulus ,Jp2,,+ q2r, be convergent (that is, absolutely convergent, since the terms are all positive), each of the component series will be absolutely convergent; but the condition is not necessary for the convergence, or the absolute convergence, of the two component series respectively.
In the series thus far considered the terms are actual numbers, or are at least regarded as constant ; but we may have a series u, + uu + u2+ .. where the successive terms are functions of a parameter z; in particular we may have a series a, + a,z+ a2z2.. arranged in powers of z. It is in view of a complete theory necessary to consider z as having the imaginary value x +iy= r(cos + i sin 4,). The two component series will then have the general terms a„,rn cos nsb and a,,rn sin n4, respectively ; accordingly each of these series will be absolutely convergent for any value whatever of b, provided the series with the general term anre be absolutely convergent. Moreover, the series, if thus absolutely convergent for any particular value R of r, will be absolutely convergent for any smaller value of r, that is, for any value of x+ iy having a modulus not exceeding R; or, representing as usual x+iy by the point whose rectangular coordinates are x, y, the series will be absolutely convergent for any point whatever inside or on the circumference of the circle having the origin for centre and its radius = R. The origin is of course an arbitrary point. Or, what is the same thing, instead of a series in powers of z, we may consider a series in powers of z - c (where c is a given imaginary value = + Pi). Starting from the series, we may within the aforesaid limit of absolute convergency consider the series as the definition of a function of the variable z ; in particular the series may be absolutely convergent for every finite value of the modulus, and we have then a function defined for every finite value whatever x+ iy of the variable. Conversely, starting from a given function of the variable, we may inquire under what conditions it admits of expansion in a series of powers of z (or z - c), and seek to determine the expansion of the function in a series of this form. But in all this, however, we are travelling out of the theory of series into the general theory of functions.
. 23. Considering the modulus r as a given quantityand the several powers of r as included in the coefficients, the component series are of the forms a, + a,cos 4, + a2cos 215+ .. and a,sin4,+ a2sin29b + .. respectively. The theory of these trigonometrical or multiple sine and cosine series, and of the development, under proper conditions, of an arbitrary function in series of these forms, constitutes an important and interesting branch of analysis.
SUM = (k( 1). For instance, let the series be z + + , nishes to zero we have - log e and (1 - e) + 1(1 - , the limit log 2 =1 –7,-1+ -- ... As a second example, consider the series 1 + z + z2 . , which for values of z between the limits ± 1 (both limits excluded) =1 - z 1– • For z = +1, the series is divergent and has no sum ; but for • z= I – as e diminishes to zero we have e-1 and 1 + (1 – E) + (1 – . . , each positive and increasing without limit ; for z = –1 the series is divergent and has no sum ; the equation 21 c - =1 – (1 – e) + (1 – 02... is true for any positive value of e however small, but not for the value e = O.
The following memoirs and works may be consulted : - Cauchy, Cours d'Analyse de l'Ecole Polyteehnique - part i., Analyse Alyebrique, 8vo, Paris, 1821 ; Abel, "Untersuchungen fiber die Reihe 1 +Tx . . ," in Crelle's Journ. de Math., vol. i. (1826) pp. 211-239, and CEuvres (French trans.), vol. i. ; De Morgan, Treatise on the DilTerential and Integral Calculus, 8vo, London, 1842 ; Id., "On Divergent Series and various Points of Analysis connected with them" (1844), in Camb. Phil. Trans., vol. viii. (1849), and other memoirs in Crumb. Phil. Trans. ; Bertrand, "Regles sur la Convergence des Series," in Liouv. Journ. de Math., vol. vii. (1842) pp. 35-54; Cayley, "On the Inverse Elliptic Functions," Camb. Math. Journ., vol. iv. (1845) pp. 257-277, and "Memoir() sur les Fonctions doublement periodiques," in Liouv. Jaunt. de Math., vol. x. (1845) pp. 385.420 (as to the boundary for a doubly infinite series); Riemann, "Ueber die Darstellbarkeit einer Function dnrch eine trimmometrische Reihe," in Cott. Ahh., vol. xiii. (1854), and Werke,bLeipsic, 1876, pp. 213-253 (contains an account of preceding researches by Euler, D'Alembert, Fourier, Lejeune-Dirichlet, he.) ; Catalan, Traite Elementaire des Series, 8vo, Paris, 1860 ; Boole, Treatise on the Calculus of Finite Differ- ences, 2d ed. by Moulton, 8vo, London, 1872. (A. C.)