Page:The New International Encyclopædia 1st ed. v. 17.djvu/881

SEBI. 80t SERIES. reduced to less than 300. Consult McGee, The Siri Indians (Washington, 18'J9). SERICITE (from Lat. serkum. silk, from Gk. cjipiKds, si'iil^-os, silky, seric, from il^P, <!>'<■. C'liinanian) . A fine scaly variety of mnseovite, cliaraotorized by a silky lustre. It is found chielly near Wiesbaden, Germany. See Muscovrric. SERICITE GNEISS, or Sericite Schist. A nietaniorphie rock, comjjo.sed essentially of the hydro-niicaceous mineral sericite (q.v.) with quartz or quartz and feldspar. In some eases at least sericite sneiss has been produced by the mashing; of granite and rliyolite (q.v.) under the action of mountain-building forces. SERICULTURE. See Silkworm; Silk. SERIEMA. A bird. See Cariama and Plate of Craxks. SERIES (Lat. xerics. row, succession, from serere, to bind; connected with (ik. clpeiv, eirein, Skt. sa, to bind). In mathematics, a succession of terms formed according to some coinnmn law ; e.g.: (1) in the series 1. 3, 5, 7.... each term is formed from the preceding Iiy adding 2; (2) in 3, 9, 27, 81, ...by multiidying by 3. A series in which each term after the first is formed by adding a constant to the preceding term is called an arithmetic series or progression; e.g. series (1) above. A series in which cacli term after the first is found l)y multiplying the preceding term by a constant is called a ficomelric series or pro- gression; e.g. series (2) above. Any term *„ of an arithmetic series is given l)y the formula ?„ = a+ i» — Drf, in which a is the first term, d the common dift'erence, and « the numlier of terms. The sum of n terms is given by the formula s := - (n + 1), I being the last term. In geometric series the corresponding formulas are feet, in the third •,'( feel, and so on indefinitely, 3 whence for the whole distance a„ = j -r or feet. IJecurriuj' decimals nuiy al.to be regarded to form an infinite series, and expressed us u fruc- tion by means of the formula »„ «n /— A series the reciprocal of whose terms form an arithmetic .series is called a hiirmonir series or progression. Hence any term ma}' be found by applying the formulas of arithmetic series to the reciprocals of its terms. Although the above are the chief series treated in elementary algebra, there is an unlimited num- ber of kinds. E.g. a type to which considerable interest is attached is the arithmetico-geometrie series, in which the coefficients are in arithmetic series and the variable in geometric series. E.g. 1 + 2a7 + 3x' + ... (» — 1) X"-' + nx"~K If the number of terms in a series is unlimited, it is called an infinite series. The general or »th term in such a series and the sum of n terms, n being indefinitely great, may or may not be determi- nate. Infinite series in which the values of 1^ and s„ (» = 00 ) are indeterminate are of little value, but those in which a limit for .<„ can be found are important. Thus in an infinite geoniet- ric series whose ratio is less than 1, s^=-^^,- E.g. to find the sum of the distances traveled by an clastic ball which falls 2 feet and bounds 1 foot and continues indefinitely to rebound one- half the distance fallen. The distance traveled in the first vibration is 3 feet, in the second IVa E.g. . (i()G .. = {'g + xJff + — 1 where a= {'g and r= ,'j. Therefore s„ = . f = 3. An infinite series in which n^, as n in- creases indelinilely, has a finite limit Is called a eunicrrienl series, otherwise a iliirrfient series. A series in which the sum is finite, hut takes alternate values as » increases, as in 1 — 1+1 — 1 + . . . , is called an osciltdling scries. The ability to determine what series are con- vergent and to determine the limit of n^ evi- dently conditions the utility of any series for the purpose of pure and applied mathematics. Thus the trigonometric functions .sinr = i — ai "' if! V + 4! series e"^ ■-l+x+^,+ ., the exponential , and the logarith- •ries log(l + ') = J — 2 + 3 are avail- able for those values only of the variables which render the series convergent. A knowledge of elementary series is very old, the Pytliagoreans ( n.f. ri.'iO) having treated them quite compreliensively. (See Number.) Euclid (C.300 B.C.) used geometric series, and infinite convergent series of the geometric type appear frequently in the works of Archimedes (c.280 B.C.). Among the Hindus. Aryabhatta. Ilrahnia- gupta, and Hhaskara treated aritlimetic series, and Bliaskara discus.sed geometric series. The Aralis did little to advance the subject and the Europeans up to the sixteenth century had made no further progress. Saint-Vincent ( l.')S4-llJ()7 1 and Jlercator (c.l()20S7) developed the series for log(l+a!), and Gregory (HUiS) those for tan ~'ar, sina-, cosj. secj, escj. The terms convergent and divergent a]i])ear in the writings of Gregory. The theory of infinite series may be said to begin with Newton and Leibnitz, and to have been further advanced by Euler. In 1812 Gauss published his celelirated memoir on the hypergeometric series (name due to Pfafft, which has since occupied the attention of .lacobi. Kummer. Sclnvarz, Cayley. Gour- sat, and numerous others. Cauchy (1821) may be considered the founder of the theory of convergence and divergence of series. lie advanced the theory of power series by his expansion of a coni]ilc- functii>n in such a form. .bel was the next important contrilmtor. and he corrected certain of C'auchy's conclusions. Gen- eral criteria liegan with Kummer (183.5). and have been studied by I'^isenstcin (1847), Weier- strass in his various contriliulion~ to the theory of functions, nini (ISfiTl. Dii Bois-Reymond (1873). and many others. Pringsheim'^ (from ISSO) memoirs present the most complete general theorv.