SERIES. 



SKIUKANT. 



UB 



Thu miMt lead to a finite expression for the nrie, and U fre<iu' :>t !y 

 the shortest WT of obtaining it 

 4. Lett*v-' -oos+ V-l . sin land fa- = a *<,* + 0,3- + . . . . 



^) } 



o + . 



the serif. resembling those In (i>. Further varieties will appear in 

 TAYLOR'S TUOBtt. 



5. If f(0) + fHl).:r + <2).:r+ . . . .1* I* development of a per- 

 fectly eootiBOOu* function of r, and if m be function which never 

 become* infinite for any real value of n, positive or negative ; then the 

 MOM function may be also developed into -f^-l). *-'-*( -2). *- 

 -f(-S). "-.... (D.C.,p.660). 



. If T vr=o + a,T + o r t i 4- ---- and if fu p v . . . p. be the *th 

 KOOTS of unity, then (U. C, i>. 319) 



. - ' 



and so on. Also if we make p,, p,, io., the nth root* of 1, and use 

 the aame result*, only altering the multipliers into p,"-', *c., p, Jm ~ 3 , 

 ,', . we hare the sum* of the same series with the terms alternately 

 positive and negative. 



7. One of the simplest modes of actually finding a finite expression 

 for a series, finite or infinite, the coefficients of which are values of a 

 rational ami integral function, is the continual multiplication by 1 .<, 

 which must at last produce a finite expression. It must be remem- 

 bered that multiplication by 1 x may be performed by letting the 

 first coefficient remain, and diminishing every other coefficient by its 

 predecessor. Thus a + b*+cx* + ex* + .... multiplied by 1 x gives 

 a + (6- a) x + (c b)z? + (e c)a? + . And a finite series must be sup- 

 posed to be continued aJ injinitum with vanishing coefficients. For 

 example, it is required to find a finite expression f or 1 s + 2V + 3V + . . . ; 

 write this as in the first line, and make successive multiplications by 

 1 jc, as in the following lines : 



1+ 8* + 27* + 64* + 126^ + . 

 61:r*+. 



1 + 6*+ 6**+ 6* + 

 1 + 4*+ *"+ 0** + 



After four multiph'cations, then, by 1 .r, the scries becomes 1 + 4. 

 ' + .I s ; whence it* value is 



(I) 4 



Independently of the modes of deriving series obtained from 

 Taylor's Theorem, and of which we are to speak elsewfcere, there are 

 two modes of forming them which deserve attention. The first 

 depends upon the numbers called the differences of nothing [Norm v. 

 DIFFERENCES or] ; the second on those called NUMBERS OF BERNOULLI. 



By the first of these any function of * can be expanded in powers 

 of x. 



/, =/!+/(! + A) 0. * + /(!+ A)0. 



Hen 



n.VTKl.N 



re /(I + ' A) 0" is a symbol of tho calculus of operations [OrE 

 >], which 



which expanded is 

 .0- 



A"0" 



it being unnecessary to go further, because A"0" = whenever m is 

 greater than a. (I). C., 307.) 



The numbers of Bernoulli occur first in the development ol 

 (t !)'->, a series the importance of which can only be estimated bj 

 its use in SUMMATION. Taking the numbers from the article cit 

 above, or making 



I 1 1 



6 



80 



B ~ 42 



B ' ~ 30 ' 



B,* 







SB,* 



tJ 



[o] 



where [u] means 1.2.3 n-l.n. 



We shall now give a number of series which are not of very 

 frequent use, but which may sometimes be sought in a work of refer 

 nee. Under the last predicament wa can never suppose that any ol 



the following development* would com* : (1 + *) , t , a- , log (1 + .r), 

 og r j; , sin x, or cos jc. Some terms are given, and .... is 

 omitted to save room. 



3? 2* 17J? 62** 1S81. 11 

 r + 3 + 16 



~c ~ 3 ~ "i3 ~ 015 ~ 4725 



* 6*' 61j 877.** 

 1 + 2 + 24 "*" 720 "*" 8064 



1 x 7x* 

 C0lwoil:= * + 6 + 360 + 



1 x> 1 .3 * 1.8.6 x* 

 su>-'j- = *+ 3-3- 



315 28~85 155925 



83555 



+ 604800 



j J.45 + 2 . 4 . 6 7 



tan -' 



log 



log 

 log 



x 



sin j- 

 1 



COS X 



tan x 



*_ 

 6 



f- 

 "2 



f! 



3 



r 180 



3* 



+ 12 

 + 90 



87800 



_*L 



' 2835 



f! 



45 + 2520 



62J* a* 



--::.> + 2700 + 





81 a-'" 



107775 



These logarithms are the Naperian logarithms, as is always the case 

 in analysis, unless the contrary be expressed (as it is usual to say, but 

 it really never happens) : 



log(l + 



2 ~ 12 "*" 24 ~ 720 "" 160 ~ 60480 



AVe must again remind the reader that the symbol + , ic., or , &c., 

 in throughout omitted to save room. 



There is one property of series which deserves particular notice 

 ( I '. C., pp. 226 and 649) as creating a most remarkable distinction 

 between those which have all their terms positive, and those which 

 have them alternately positive and negative. The former, even if the 

 terms diminish without limit, are not necessarily convergent ; thus 

 l+2~'+3~ 1 + .... is divergent. But if the terms be alternately 

 positive and negative, and diminishing without limit, the series is 

 always convergent, and tho error made by stopping at any term is less 

 than the first of the terms thrown away. And the most remarkable 

 part of the property is that this last iu true, even if the series become 

 divergent, by having its terms increasing without limit, instead of 

 diminishing; so that if the terms diminish for a time, and then 

 to increase, the portion of the series during which the diuii: 

 takes place may be made use of in approximating to its aritliii 

 value. That is, if A,, A,, be all positive quantities, and if tin- inliniU: 

 series A, A, + A, .... be carried as far as A, , the error is less than 

 A. + i, whatever the law of the terms may be, or however rapidly 

 they may afterwards increase. Let us take, for instance, 



2.3 

 ~ 



2.8.4 2.8. 4. 5 



Let x be ever so great, the rapid increase of the numerators must .-till 

 make this series ultimately divergent. Nevertheless, if x be con- 

 siderable, tho first terms diminish so rapidly that, with the aid of tho 

 above theorem, a good approximation may be made to the arithmetical 

 value of the function from which the series was derived. Let x= 100, 

 whence the series becomes 



1 _ -02 + -0006 '000024 + "00000120 - .... 



A f ter tho hundredth term the terms will begin to increase, and more 

 and more rapidly ; but the theorem enables us, when x = 100, to make 

 the following assertions ; first, 1 is too great, but not by so much as 

 02; 1 '02, or -98, is too small, but not by so much as '0006; 

 98 + -0006, or -9806, is too great, but not by so much as -000024 ; -9806 

 -000024, or -980676, is too small, but not by so much as -00000120 ; 

 980576 + -00000120, or -9805772, is too great, but not by so much as 

 the next term ; and so on. 



SKIUEANT, or SERGEANT (Senitni). This term, in its original 

 signification, has long become obsolete. It would however be d : 

 to trace the connection between the different officers to whom the 

 term is now applied without going back to their common type. 



The term servieus and serviciuin appear to have been applied at first 

 to all servants of the public, or of the crown, as the head of the state, 

 and to the service rendered by them as an acknowledgment or render 

 for the lands held by such service. Rent paid by a tenant to his land- 

 lord u still distinguished by the name of rent-service [ UKXT] from 

 other annual payments charged upon land, &c. The word " Serjeant" 

 comes to us from " sergent, into which tho French had modified the 

 Latin " semens." The word serjeanty, in French " sergenterie," was 



In these series the angle 1 len than a right angle, poaltirc or negative. 



