SERGIUS, I: b. Syria, about 630; d. Rome 701. He succeeded Conon in 687. He opposed several provisions of the Quinisext Council of Constantinople in 692 and in consequence Justinian II sent to arrest him. Sergius, however, was protected by the exarch of Ravenna. He died in Rome after a pontificate of nearly 14 years.

SERGIUS II: b. Rome; d. 847. He succeeded Gregory IV in 844, and during his pontificate the Saracens from Africa ascended the Tiber, and plundered the environs of Rome.

SERGIUS III: d. 911. He was raised to the papal chair in 904, after the anti-pope Christopher had been expelled. On the strength of Luitprand's testimony, his character has been deeply calumniated. But contemporary chroniclers speak of him in the highest terms as a pious and enlightened pope. He restored the Lateran Chur, and filled the Holy See nearly seven years.

SERGIUS IV: d. 1012. He succeeded John XIX, in 1009, and during his pontificate, and as a result of his exhortations, the Italian princes united to drive out the Saracens from the country. It was in his time also that the Normans began to enter Italy.

SERI (sā-re') INDIANS (Opata, “spry»), an exceedingly primitive tribe of North American Indians, speaking a distinct language and living on Tiburon Island, in the Gulf of California, and the adjacent mainland of Sonora, Mexico. Although known since 1540, when some of Coronado's men visited them, almost nothing was known of their character, save that they were warlike and extremely conservative, until Dr. W. J. McGee led a small expedition among them, in the interests of science in 1894 and again in 1895. The Seris are unusually tall and well-built; they wear little clothing and subsist chiefly on turtles, water-fowl, fish and other food of the sea, eked out with the vegetal and animal products of the mainland desert. Their houses are flimsy bowers of cactus and shrubbery, sometimes rudely shingled with turtle shells and sponges. Bows and arrows are habitually used as weapons, harpoons of cane are employed for taking turtles, and unworked bowlders or cobbles form a ready means for serving meat, crushing bones, mulling seeds, etc. They make graceful balsas of canes lashed together for use in navigating the strait between their island and the mainland; manufacture a very light pottery and some basketry. Shells are used for cups and to some extent for implements. The modern Seris are loosely organized in a number of maternal groups or clans, which are notable for the prominence given to mother-right in marriage and for some other customs. Polygamy prevails. These savages manifest an implacable hatred toward aliens, whether Caucasian or Indian, and the shedding of alien blood is regarded as their highest virtue. Two centuries ago the population of the tribe was estimated at several thousands, but it has been reduced by almost constant warfare, so that now it is barely 350. Consult McGee, The Seri Indians; 17th Report of the Bureau of American Ethnology, Washington, D. C.

SERICULTURE. See SILK AND SILK

INDUSTRY.

SERIES. A series is a succession of numbers or terms, usually proceeding according to some definite law. The law of the series may either define how a given term is derived from those that precede it, or it may define a term as a function of the number denoting the position which it occupies in the series. Thus, the series 1, 2, 3, 5, 8, 13, . . . is such that each term after the first two is obtained by adding together the two terms immediately preceding it, and the law of the series might be expressed by the formula un+ı = un-1 + un, in which un-1, un, un+1 are any three consecutive terms. On the other hand, the series 1, 3, 1, 4, ... may be defined by the law that the nth term of the series is Each term is then a function of the num

n

2n -1' ber n which determines its position in the series. One and the same series may frequently be defined in both ways. Thus, the series, 1, 1, 1, 1, . . . is such that each term may be obtained from the preceding one by multiplying it by. It may also 1 be defined by the formula as the expres 2n-1 sion for the nth term. A series is not uniquely determined when a number of its terms are given. In fact, an infinity of series may be found which include as terms any finite number of arbitrarily assigned numbers. In such a case the simplest law that can be found is naturally to be taken as the law of the series. Important use is made of this principle in the physical sciences. Series of numbers frequently arise in connection with experimental observations, which evidently proceed according to some definite law, and it is often of great interest to determine the nature of the law. Such series occur, for example, in the theory of spectrum analysis. Balmer's formula n2 in which c is a certain constant, detern2. -4' mines, for different integer values of n, the wave-lengths of a long series of lines in the hydrogen spectrum. A variety of formulas have been obtained for other spectra. A very instructive account of the methods used in determining such series is given in Kayser's Handbuch der Spectroscopie, Vol. II, pp. 503-573.

A series is said to be finite or infinite according as the number of its terms is limited or unlimited.

FINITE SERIES.

The law of the series being given, the most important problems relating to a finite series are the determination of the term occupying a given position and finding the sum of the series. The following cases are those of chief interest.

Arithmetic Series. This series is a, a + d, a+2d, . . ., l, so that the difference between any two consecutive terms is constant and equal to d. The nth term is evidently obtained by adding d n-1 times to a and hence = a+ (n-1)d. If S denote the sum of the series, then Sa (a + d) + (a + 2d) +. . . + 1. By writing the terms in reverse order, the sum may also be expressed in the form

S=1+ (1 − d) + (1 − 2d) + . . . + a. The addition of these two equations gives 25= (a + 1) + ( a + 1) + to n terms, whence S=1n (a+1)=n[a + (n − 1)d]. In particular the sum of the first n integers 1, 2, 3, . . ., n is

in (n+1), and the sum of the odd integers 1, 3, 5, 2n-1 is n2.

Higher Arithmetic Series. The general expression a+ (n−1)d for the nth term of an arithmetic series is an integral function of the first degree in n. A more general series may be formed in which the expression for the nth term is an integral function of degree s in the n. Such a series is called a higher arithmetic series of order s. Let ai, dɑ2, ɑз, . . represent the series. A new series b1, b2, b3, is formed from this by taking the differences of consecutive terms, so that b1-as-a, bɔ ɑs d2, etc. This is called the series of first differences. Let a series C1, C2, . be formed from b1, b2, . . . in like manner. These are the second differences. Let di, ei,... be the first terms in the series of third, fourth, . . . differences respectively. Since an is a polynomial of degree s in n and anti is the same polynomial in (n+1), it follows that bn, which equals anti-an, is of degree s-1 in n; and, finally, that the nth term of the series of (s-1)th differences is of degree one in n; that is, the last series of differences is a simple arithmetic series. The series of sth powers of the natural number 1, 28, 38, ..., ns is the simplest example of a higher arithmetic series of order s.

The nth term of the series a, a, as... can readily be expressed in terms of n, b1, C1, namely, an=a1+(n-1)b1 +'

+

(n-1) (n-2) c

1.2 (n − 1) (n — 2) (n—3) 1.2.3

the last term of the formula being of degree s in n. With fractional values of n this formula is much used for the purpose of interpolating between the terms of a given series numbers which shall follow the same general law of variation. For example, suppose it be required to find log 91.43, given log 911.9590414, log 92 1.9637878, log 931.9684829, log 94: 1.9731279. The required number is intermediate between log 91 and log 92, which are the first and second terms of the series having log (90+ n) for its nth term. As the number 91.43 corresponds to n=1.43, the value of log 91.43 may be calculated approximately by assuming that for a small number of terms the series of logarithms coincides very nearly with a higher arithmetic series of order s, the closeness of the approximation increasing with higher values of s. The first terms of the successive series of differences are b1 =.0047464, C1.0000513, d1.0000012, etc. First try s=1, that is, use only two terms of the formula for an. This gives log 91.43 1.9590414+ .43 (.0047464)=1.9610824, which is correct to only five decimals. With s=2, log 91.43 1.9610886, which is less than the true value by one unit in the seventh place of decimals. With s 3, that is, by using four terms of the formula for an, the value is obtained correctly to seven decimal places.

Geometric Series.- This is composed of the terms, a, ax, ax3, ax3, . . . Each term is formed from the one preceding by multiplying it by x, so that the nth term is axn-1. If S denote the sum of n terms of the series, then S= a + ax + ax2 + . . . + axn−1.

VOL. 24-38

593

1,

Recurring Series.-A more general form of series is the arithmetico-geometric series, in which the nth term is of the form (a+bn)xn−1 It reduces to the arithmetic series when and to the geometric series when b=0. This in turn is a special case of a recurring series, which is defined as follows: Let as, arx, a3x2, anxn-1 be the terms of such a series. If r+1 successive coefficients a; are connected by a homogeneous linear relation of the form an + pian-1 + P2an-2 + ... + pran-r=0, in which P1, P2,..., Pr are given numbers and n>r, the series is called a recurring series of order r. The sum of n terms of a recurring series can always be found. The result is a rational fraction in . The method of summation may The be illustrated by the series of order 2. defining relation is

an + pian-1 + P2an—2=0. Let S a1 + A2x + α3x2 +. +anxn-1. Multiply by 1+pix + pax2. This gives (1+pix +Þ2x2)S=a1 +(a2+pia1)x

+(a3+ pia2+ P2a1)x2+(a1+ piɑ3+ prɑ2)x3
+...+(an + pian−1 + pran-2)xn−1
+(anpi + an−1Þ2)xn + anÞ2xn+1.

Since as + pias−1 + pras¬2=0, the above expression reduces to

a1 +(a2+ pia1)x+(anpı+ an−1p2)xn+anр2x2+1, and hence

Convergent Series. Let u1, U2, un, be the terms of an infinite series, and let Sn denote the sum of the first n terms. If Sn approaches a definite, finite limit as n increases to infinity, the given series is said to be convergent. If, on the other hand, Sn increases without limit, the series is divergent. If Sn does not approach any limit, the series is called indeterminate, as in the case of the series 1−1 +1−1 +1-..., for which, Sn takes the values 1 and 0 alternately as n increases. To illustrate these definitions, consider the infinite geometric series a + ax + ax2 + . . . + axn−1 ... By the formula given above for the a(1—xn) sum of a geometric series Sn= 1-x

=

If x is numerically less than unity, rn diminishes to zero as n increases to infinity, and hence Sn

is called the generating function of the series. The series and its generating function are equal for all values of x for which the series is convergent, that is, for values of r numerically less than 1. But if r is numerically greater than 1, Sn approaches infinity as n increases without limit. The series is then divergent and is no longer equal to the generating function. Similarly, with a recurring series, if its terms approach zero as a limit with increasing n, the terms anxn, an−1xn, anxn+1 in the above formula for the sum of n terms of a series of order 2 approach zero and hence a1 + (a2 + pia1)x Sn has for limit S A like 1 + Þ1x + Þ2x2 result holds for a recurring series of any order.

A series can represent its generating function only for such values of the variable (or variables) on which it depends as will make it convergent, and hence the question of the convergence of a series is of paramount importance. Suppose at first that the terms of the series are positive. Then, if the ratio un+1:un becomes and remains less than a proper fraction k as n increases without limit, the series u1+u2+ +un+ is convergent. For in that case Un+1<kun, Un+2< kun+i <k2un, Un+3< kun+2<k3un, and so on. Hence un +un+1 + un+2 + Un+3 + un ( 1 + k + k2 + k3 +...). But as k is a positive number less than 1, the infinite geometric series 1 1 + k + k2 + ... is convergent and equals1-k' Accordingly un + Un+1 + which is less than the finite quantity is a convergent series. If to this portion of the series the finite sum 1 + 12 + . +un-1 be added, the result is finite and the given series is, therefore, convergent. While the condition just given is sufficient to ensure the convergence of a series it is not necessary. Thus, if a and b are such that 0<a<b<1, then the series a + b2 +a3 + b2 + a + ... is convergent, since it consists of the sum of the two convergent geometric series a + a3 + . . . and b2 + ba + · But un+1:un alternates between large and small values as n increases and does not, therefore, remain less than a proper fraction.

...

un

1 k'

If some of the terms of a given series are negative, the series is convergent provided that the series obtained by changing the signs of all the negative terms is convergent. For example, the series 1 — k + k2 — k3 + k1— . . ., in which k is a positive number, is less than the series 1 + k + k2 + k3 + . . . and is, therefore, convergent if k<1. It follows from the preceding considerations that any series is convergent if the ratio n+1:un has a limit numerically less than 1 as n increases to infinity. Thus, for the logarithmic series log (1 + x) = x − x2 +} x3 Xn xn+1 the numerical value of n n + 1 xn+1 xn un+1:un is the same as that of n+1 n n + 1 The limit as n increases to infinity is x. The series will accordingly be convergent if x is numerically less than 1. For the exponential series x2 x3 2! 3!

+

:

=

ex=1+x+ + + + +

xn

n!

xn-1

n

x.

A series of positive and negative terms is said to be absolutely convergent if the series obtained by making all of the terms positive is convergent. When the second series is divergent it sometimes happens that the first series is convergent, in which case it is called conditionally convergent, because the value of the sum is conditioned on the arrangement of the terms. For example, the logarithmic series is absolutely convergent for all values of r numerically less than 1. If x=1, it becomes 1— }} + } − 1 + } which is convergent and equals log 2. But the series 1 + 1 + 1 + · ·, obtained by making all the terms positive is divergent. A conditionally convergent series has the peculiar property that by changing the sequence of the terms the sum of n terms may be made to approach any desired limit as n increases to infinity, as may be exemplified by the preceding series. For let c be any number. Select in order as many of the positive terms of 1 ++... as will be sufficient to make a sum greater than c. Then select just enough negative terms so that when added to these the sum shall become less than c. Then add more positive terms until the sum becomes greater than c, and so continue. This process is possible, since the positive or the negative terms alone form a diverging series. With such an arrangement of terms the sum may be made to approach as near to c as desired.

Uses of Series.- Series are of the utmost value as affording a means for the expression of any given function. In particular, functions which are defined as solutions of differential equations are most frequently not expressible in any other form. In the theory of functions as developed by Weierstrass the convergent power-series is made the fundamental instrument of investigation, and from the properties of its defining series the nature of a function is determined. (See COMPLEX VARIABLE, THEORY OF FUNCTIONS OF). The expression of a function in the form of a series is of especial importance when it is desired to calculate its numerical value. It is by this means that various numerical tables, such as the logarithmic and trigonometric, are calculated.

Taylor's series: f(x)=f(a) + f'(a) (x− a)+ 1 —11 ƒ′′ (a) (x − a)2 + ·· + = f(n) (a) (x − a)" + . . . is the formula most generally useful for the purpose of expanding any given function f(x) in ascending powers of x- -a, a being a given constant so chosen that the functions and its derivatives f'(x), ƒ''(x), . . ., f(n)(x), ... are finite and continuous for values of sufficiently near to a. Another form for Taylor's series, and the one to which the name is more frequently applied, is obtained by writing hin the place of x- a so that x=a+h. The above equation then takes the form f(a + h)=f(a) + f'(a)h

series by putting a=0. The formula is of course inapplicable if the given function or any of its derivatives are not finite when r=0.

For the purposes of numerical calculation it is essential to know how close an approximation to the true value of the function may be obtained by using a given number of terms in Taylor's series and discarding the rest. If Sn denote the sum of the first n terms of the series and Rn the remainder of the series, then f(x)=Sn+ Rn and the amount of error introduced by using Sn as the approximate value of f(r) is equal to the remainder Rn. An expression for the remainder in Taylor's series (second equation) was obtained by Lagrange in the 1

form Rn ¡f(n) (a + 0h)hn, in which @ denotes a

n

mals. For the remainder is R2: = the first factor of which differs slightly from while h2 has no significant figure before the eleventh decimal place.

1

2V2

The cases in which Lagrange's formula is practicable for determining the limit of error are comparatively few in number. A convenient limit of error may frequently be found by comparing the given series with a simpler one of less rapid convergence, the limit of error for which can be determined. In particular, a series u — U2 + Uz — Us + ... of alternately positive and negative terms which are numerically decreasing toward zero as a limit is convergent, and the error in taking only n terms is less than the next succeeding term un+1, but greater than un +1 un+2. For the series may be written (u1 ·U2) + (uz 4)+, which is a sum of positive terms and, therefore, increases, with the number of terms, toward a finite limit, or becomes indefinitely large. But the series may also be written u-(u-U3) —(U4 — U5) which all the terms after the first are negative and hence the sum is less than u1. The numerical value of Rn is unti un+2+un+3which,

in

by a repetition of the preceding argument, is greater than Unti un +2 and less than un+1. Thus, for the logarithmic series mentioned above, the error in stopping with the nth term is xn+1 1 less than (if x is positive). For x the value of log (1+x) is given by Sn correct

n + 1

10

595

1

the sum of which is 2(2n+1)(2z + 1)2n+1(z2 + %)" The error in neglecting Rn is accordingly less than the value of this expression.

The importance for numerical computation of having a series of sufficiently rapid convergence is illustrated by the formula for π. Gregory's series is 1-+-+... which converges so slowly that according to Newton it would require 5,000,000,000 terms and 1,000 years' time to calculate the value of to 20 decimals. On the other hand, by means of Machin's formula, which converges very rapidly, Shanks has computed the value of to 707 decimal places.

Divergent Series. It was supposed by the earlier mathematicians that divergent series were entirely useless. There is, however, a class of such series that have been found not only useful but necessary for the purposes of approximate computation. These have been employed for a long time by astronomers even after their divergence had been proved, their use being justified by the close agreement between calculation and observation. The series in question have the property that the terms at first decrease in numerical value until a certain minimum term is reached, after which they increase without limit. The series is converging as far as its minimum term, beyond which it is diverging. The converging portion of the series frequently affords a very good approximation to the value of the given function. The

beyond a certain term, the sum of the converging portion gives an approximation having an error less than the numerical value of the last term used. This approximation is so good and the convergence so rapid that for values of x as small as 3 this series is more convenient to use than the first one and gives a value correct to eight decimals, the first 11 terms being convergent, while for r = 10 the first 100 terms are convergent and give a value which is correct to at least 110 decimal places.

The theory of divergent series has recently been receiving considerable attention, chiefly from the French mathematicians. In particular, the problem of determining in how far the divergent series may validly be used as a purely formal representation of a function has been studied by Poincaré, who has developed some important and far-reaching results. An interesting account of the present state of the subject is given by Borel in his 'Leçons sur les séries divergentes' (Paris 1901).

Series of Functions; Multiple Series. It is often important to express a function as a series the terms of which are given functions such as trigonometric or Bessel functions. Such series are of great service in mathematical physics, especially the Fourier series of sines and cosines of multiple angles, which is much used for representing functions having a periodic character. This series was first employed for that purpose by Fourier (1822) in his 'Théorie analytique de chaleur.)

Un

In a simple series u1, uz,... the general term is a function of a single integer n defining its position in the series. A double series is one in which the general term um, n is a function of two independent integers m, n. A multiple series is one in which the general term is a function of several integers m, n, p, Such series are of fundamental importance in the study of various classes of functions such as the elliptic functions and the multiple theta functions.

History and Literature. Some of the properties of arithmetic and geometric series were known in very ancient times, as the oldest manuscript in existence, the Rhind papyrus, which is believed to be a copy of a work dating back 3,000 or more years before Christ, gives an account of some problems which are solved by means of these series. There is considerable ground for supposing that the general formula for the sum of an arithmetic series was known at that time, although it does not occur explicitly in mathematical literature until the time of Archimedes (287-212 B.C.). The formula for the sum of a geometric series was known to the Greeks as it is given by Euclid (about 300 B.C.), while the summation of the infinite series 1+1+(4) + (})° + ... was effected by Archimedes. The general expression for the sum of an infinite geometric series was first given by Vieta (1540-1603 A.D.). A formula for the sum of the squares of the positive integers was given by Archimedes, and for their cubes by Nikomachus (about 100 A.D.). Except for these two particular cases no knowledge of the summation of higher arithmetic series is apparent before the 16th century. The subject as we now know it was developed chiefly in the following century and it took its present form in the 'Ars conjectandi' of Jacob Bernoulli (1713).

>

The name "arithmetic series of higher order» was given by Leibnitz. Newton was the first to recognize the importance of infinite series as an instrument of mathematical investigation. His results were published in 1669 (not printed until 1704) in a work entitled 'Analysis per Aequationes Numero Terminorum Infinitas. A systematic treatment of series was given by Euler in his 'Introductio in analysin infinitorum (1748). The principle that an infinite series cannot be safely employed unless it is convergent was first given in this work. Gauss (1813) inaugurated the modern epoch in the theory of infinite series in his memoir, 'Disquisitiones generales circa seriem infinitam, ab a(a + 1)b(b + 1) x2+... He insist 1.2.c.(c+1)

1.c

1+ x + ed on rigorous tests for convergence and was first to give definite criteria for convergence and divergence. In the Cours d'analyse de l'Ecole Polytechnique' (Paris 1821), Cauchy extended the theory of power-series to the expansion of functions of a complex variable and gave the important theorem that a series is convergent if the limit of uVis numerically less than 1 when n increases to infinity. Further contributions, including additional criteria for convergence and a development of the principle of continuity of series, were made by Abel (1826) in his 'Recherches sur la série,

m

1 + 1 x +

Euvres complètes,' Vol. I, p. 223). This celebrated memoir is a model of scientific accuracy and comprehensiveness, and exerted an important influence on the growing spirit of rigorous and critical investigation.

Among modern works on series the following may be mentioned: Chrystal's 'Algebra' (Edinburgh 1886), chaps. xx, xxvi-xxxi; Osgood's Introduction to infinite series' (Harvard University Publications 1897); Godefroy's 'Théorie élémentaire des séries' (Paris 1903); Runge's Theorie und Praxis der Reihen' (Leipzig 1904); Borel's 'Leçons sur les séries à termes positifs) (Paris 1902); Hadamard's 'La série de Taylor) (Paris 1901); Jordan's 'Cours d'analyse (Paris 1893), Vol. I, chap. ; Encyclopädie der Mathematischen Wissenschaften (Leipzig 1898-1904), Vol. I, part 1, pp. 47-146.

JOHN I. HUTCHINSON, Professor of Mathematics, Cornell University.

SERIES, the term applied to the rocks laid down during an epoch. Thus we speak of the Acadian series as laid down in the Acadian epoch. Series is a subdivision of system (q.v.) just as epoch is a subdivision of period (q.v.).♦

SERIES-PARALLEL CONTROLLER.

See ELECTRICAL TERMS.

SERIN, in ornithology, a finch, Serinus hortulanus, which is closely allied to the canary. The bird's mantle and back are of a darkgrayish brown, each feather being edged with a broad strip of yellow. Its head is olive-gray and the throat and breast are of a bright gamboge-yellow, shading to white on the belly. It is four and one-half inches long. The serinfinch is noted as a lively and indefatigable singer. It is migratory and spends the summer and oftentimes the winter in Middle Europe.