822 



SERIES 



SEROUS FLUIDS 



\ + . . . to infinity has not a finite value, U not 

 convergent, although iu infinite term is zero. 

 Such a series is divergent, and cannot be gummed 

 to infinity. To prove tliis throw the KIT!*-* into 

 groups of tenim, tin' lii-i group lieing the first term, 

 tin- second the next tiro, the third the next fmir, 

 the fourth the next riijht, and mi on. Thu tin- 

 lift li group will consist of sixteen ternia, beginning 

 with ,> and ending with , 1 ,. Each of these frac- 

 tions if greater than J, or 1/2* ; so that their mini 

 is greater than sixteen times this quantity .., 

 a 4 / 1 !* or J. Hence, if we go aa far as m groups, 

 the series will lie greater than 1 + Jm. Thus by 

 taking in large enough we can make the -HIM iis 

 large as we please. The series is divergent and 

 cannot he summed. We may, however, by simply 

 changing the algebraic sign of every alternate 

 term, obtain a series which is convergent viz. 

 1 - i + J - J + . . . to infinity. That this series 

 has a finite sum may be made evident graphically 

 thus: Take AB equal to unity; this is the first 



term. Move back half-way to C ; this gives the 

 second term mimix one- half. Move forward to D, 

 where CD is one-third ; then back to E, where 

 DE is one-fourth ; and so on indefinitely. It is 

 clear that we shall ultimately oscillate through 

 diminishing ranges about some point between C 

 and B; so that the sum of this series is less than 

 1 but greater than J. The series, in fact, is the 

 Nanierian logarithm of 2, and has the value 

 69315... A series like that just given, which 

 is convergent only when the signs of the suc- 

 cessix'e terms differ according to some definite 

 rule, is usually called semi-convergent. A series 

 which converges when all its terms have the same 

 sign is said to be absolutely convergent. Sir (I. II. 

 Stokes long ago distinguished them as arriilrntnf/,> 

 and essentially convergent, a terminology whidi 

 seems in many respects superior to that in common 



11-0. 



It is important to have a test of convergency ; 

 and the most useful test is to take the ratio of two 

 consecutive terms, and consider what value this 

 ratio approaches as we take the terms higher and 

 higher. This ratio is called the ratio of con vergoncy ; 

 if it is ultimately less than unity the series is con- 

 vergent ; if greater than Qatar, divergent. This 

 test, however, gives no information when the ratio is 

 ultimately unity. As an example, consider the 

 exponential series : 



Here the ratio of the (n + 1 ) to the nth term is 

 *l(n + 1 ), which is ultimately zero, since whatever 

 value x may have n can lie taken as large as we 

 please, so that the ratio may he made smaller than 

 ariyaHsignablec|iiantity. Asiswell known, the value 

 of this series is . , where e has the \alue 271828. . . 

 l.<;.\itiniMs). Convergent Kcries are of indis 

 pcnsalile service in the calculation of logarithm* 

 and trigonometrical functions and in many im- 

 portant physical applications. Nc,t a f ew n ( their 

 properties were consequently known to the ear- 

 lier analysts; but it is to Cauchy (1827) that we 

 owe the foundation and partial development of 

 the modern theory of convergence. Diriclllet, 

 Abel, Gauss, I)e Morgan, Bertram), Kummer. Mu 

 Bois-Reymond, and others have ably supplemented 

 Canchy> work. A very complete 'introduct ion to 

 the whole subject is given in Chrystal's Algebra 

 ( vol. ii. ). There also will lie found a discussion of 

 certain parts of the subject which we can only 

 name, such as oscillating series, double series, 

 infinite products, reversion of series, and the like. 



See CIRCLE and TRIGONOMETRY for some particular 

 caseo of series. 



Serinitffar. See SRIXAOAR. 



ScrillKapatHIll (properly Sri Ranya Pata- 

 nam = ' City of Vishnu ' ), the capital of Mysore 

 state in Southern India from 1610 to 1799, is 'limit 

 on an island in the Kaveri, 10 miles N K. .,i the 

 city of Mysore, The island is three miles long and 

 one broad ; at its western end stands the fort, sin 

 rounded by strong walls of stone, and enclosing the 

 palace of Tippoo Saib and the principal mo- 

 I Outside it are the garden in which was Imilt the 

 mausoleum of Tippoo anil his father, II \.ler Ali, ami 

 Tippoo's summer palace. The fort wits licsicgcd hy 

 Lord Cornwall!* in I7!l, and again in \~'.f>. ( )'n 

 the last occasion the terms dictated hy the Mnti.-h to 

 Tippoo were very severe, A British army ap|>oared 

 liefore the walls again in 1799, and on the :td May 

 of that year the fort was stormed and Tippoo slain 

 in the vicinity of his own palace. Pop. 150,000 in 

 Tippoo s day; 32,000 in 1800; and now 10,000, 

 most of whom live at the suburb of Ganjam, the 

 ancient city being now in a very ruinous condition. 



Si-rill-diail! (Srimngam), a town in the Madras 

 Presidency, on an island in the Kaveri, 11 miles 

 W. of Tnchinopoly, with a pop. of 21,632. The 

 place is noted for its great temple of Vishnu, a 

 vast complex of halls and gopnras (colossal gate- 

 ways) built on no very regular plan, but enclos- 

 ing so large an area that most of the houses of the 

 town are within the temple walls. Notable is one 

 'hall of a 1000 columns '(960 really), 450 feet Ion" 

 hy 130 wide. 



Serjcant-nt-Arms, in the English Court of 

 Chancery, is the officer who attends upon the Lord 

 Chancellor with the mace, and who executes by 

 himself or deputies various write of process directed 

 to him in the course of a Chancery suit, such as 

 apprehending panic- who are pronounced to be in 

 contempt of the court. A similar officer attends on, 

 each House of Parliament, and arrests any person 

 ordered by the House to be arrested. 



Srjeant-nt-Lnw used to be the highest 



degree of hamster in the common law of England. 

 The degree is of great antiquity, and formerly a 

 barrister could only be appointed after lieing of 

 sixteen years' standing. Formerly, also, they hat! 

 exclusive audience in the Court of Common Pleas. 

 The proper forensic dre of MM jeants was a violet- 

 colotircd roln' with a scarlet hood, and a hlack coif, 

 represented in modern times 1,\ paidi uf silk at 

 the top of the wig. A Serjeant was appointed by a 

 writ or patent of the crown. The Chief, justice of 

 the Common Pleas recommended the Kanister to 

 t In- Lord Chancellor, who advised tlie cron. The 

 degree of Serjeant was entirely honorary, and 

 merely gave precedence over barristers : and when 

 he was ap|Hiinted he was rung out of I lie Inn of 

 Court to which he lie-longed, ami thereafter joined 

 the brotherhood of Serjeants, who formed a separate 

 community. By ancient custom tin- common-law 



judges were always admitted to th der of scr- 



jeants before sitting as judges, but this practice 

 was alMilished in 1H74. The society of Serjeants 

 Inn was dissolved not long after, and the order is 

 now extinct; a few surviving Serjeants retain the 

 title. See the article Cm K : I'nlling's tint,; ,,/' the 

 Coif( 1884) ; and Worbrydc's /://,,'/.,/ Serjetmtt-at- 

 Law (I860). 



Sermons. See PREACHINC. 



Serous Fluids, various fluids occurring in 

 the animal Imdy, are arranged by Corup-liesancz 

 under three heads : ( 1 ) Those 'which are con- 

 tained in the serous sacs of the body, as the 

 cerebro-spinal fluid, the pericardia! fluid, the peri- 

 toneal fluid, the pleural fluid, the fluid of the 



