IIECURKING SER1E.S. 



forms by fubftituting a for a, 

 which cafe the above become 



i X for (S, ex'' for y, &c. in 



S = 



_ a + (6 — a ju) 



S = -T- 



I — il. X — t X 



l-t (^ ~ '' ^') •»' + ((^ — i /i — a ») w^ 



I — /J. X — ► jr' — f .r' 



a + (i — aix)x + (c ~ b^L — av) x'' — (d — Cf* — 



Qj i» — Of) w' 



I — /* »: — V A'* — jx' — 0- .r' 



which are the feveral exprellions for the fcales of relation o' 

 two, three, or four terms, and the law of formation is fuffi- 

 ciently obvious for any other fcale. 



The fame law might have been otherwife obtained as fol- 

 lows. It is evident, from what is done above, that the fcale 

 of relation fubtrafted from unity is always equal to the de- 

 nominator of the generating feries, and the fame may alfo be 

 demonftrated on other principles. In order, therefore, to 

 find the numerator, we might alTume the propofed feries 



a +ix + cx'' + dx^+ &c. 

 _ a' + l/'x + c' .<^ + d' .V' + &c. 



I — fJiX — v;c^ — ^ x^ — &c. 



then multiplying this denominator by the propofed feries, 

 and equating the co-efficients of the produtis with thofe ot 

 the like powers of x in the numerator, we (hould find 

 a' = a 



V ^ b - a i^ 

 c^ = c — b jj. — a V 

 d<= &c. &c. 

 as in the preceding formulx. 



It is only neceuary to obferve, farther, that if the^indices 

 of X do not follow the law we have fuppofed, we muft make 

 the neceflary correftion in the general exprelTion, and if the 

 propofed feries have no power of x in it, as commonly hap- 

 pens, we muft, in the conclufion, make v = + I. 



The above formula, it will be obferved, give the fum of 

 the infinite feries. When only a certain number of terms 

 are to be fummed, different formulae are required, which we 

 fhall inveiligate, after having ihcwn how to obtain a general 

 exprellion for the nth term of fuch recurring feries. We 

 'propofe, however, in the firlt place, to illuftrate the above 

 rules by one or two examples. 



Exam. I. — Required the fum of the infinite recurring 

 feries i + 6 x + iz x'' -Y j\& x' -[■ &c. the fcale of rela- 

 tion being i + 6. 



Here a = i, 5 = 6, p = i, and v = 6, whence 



a + (b — Ofj.) X > + S * 



_ 15 -, 



I — n X — > x I — x — o ji 



the fum required. 



Exam. 2 — Required the fum of the infinite 

 feries 



63. 



1 + ^ X + 6x'^ + 1 1 A-' -f 28 ** -f 

 the fcale of relation being 2,-1, f $• 



Here 0=1,4=4,^ — 6; /*=2» » = — 

 e = 3 ; whence 



a + {b — a fj.) X + {c — b (jl — ai) _ 

 1 — ft X — V x'' — { x' 



I + (4- 2)x + (6- 8 + 



recurrmg 

 &c. 



S = 



0^-'_ 



(1 



- 2X + x' ~ 3 x' 

 + xY - 2 x" 



as required 



(I -■»•)'- 3*' 



^^^- — To find the general I crm of any propofed recurring 

 feries. 



From the preceding pan of this article it appears,, that 

 every recurring feries may be confidercd ai arifing from thc 

 devclopement or expanfion of lome rational fradion of th' 

 form 



n ■{■ bx -\- c x"- -'r </*' 4 &c. 



I — a, X ~ 



■ x' - y x> - 3 xf* — &c. 



Let us, therefore, fuppofe this fraftion to be converted 

 into the infinite recurring feries A + B .r -f- C a' -}- D x 

 + E .v' -I- &c. of which we already know how to deter- 

 mine the co-efficients, and the law of their formation. 



Now if this rational fraftion be decompofed into itb fiin- 

 ple fraftioiis by the method explained under the article Ra- 

 tional Frafiions, and each of thcfc fimple fraftions be then 

 converted into a recurring feries, it is evident that the fu.m 

 of all thefe feries ought to be equal to the original feries 



A -f- B ;c -J- C ;t:' -f D ;r' -h E *• + &C. 



Now each of thefe partial fraftions being of the form 



A' 



— — , the feries thence arifinc; will have the form 



A' + A'r.x- + A'r^«' + A' )•■':«•' 



A' 



of which A r" x" is the general term. Hence, the feveral 

 feries arifing from the partial fraftions may be fuppofed 

 to be 



and fince the fum of thefe feries is equal to the original 

 one propofed, we know that the co-efficients of the like 

 powers of x are alfo equal, whence we have 



whence it appears that the co-efficient of any term x" of the 

 recurring feries is equal to the fum of the co-efficients of the 

 fame power of x, which arife from expanding the feveral 

 fimple fraftions into which the given fraftion is decompofed ; 

 and this co-efficient is always equal to the fum of each of the 

 numerators of the feveral fimple fraftions, multiplied into 

 the «th power of the correfponding value of r in the deno- 

 minator of the fame fraftion, at Icaft while the denominator 

 contains no equal faftor. But if, among the partial fraftions, 



A' 

 there is any one of the form . — , the general term of 



this will be (n + 1) A 

 (u + i) (h -I- 2) 



{l-rxY 

 of 



A 



and term is 



I . 2 



the general term is 



(n+ 1) (« + 2){n + 3) 



(l-rxy 

 A r" x"; and univerfally 



the general 

 A 



{l-rxy 



{n + i-l) 



1.2.3... (^-i) 



we may, therefore, in all cafes wherein the generating 

 fraftion of the original feries admits of a rational decompo- 

 fition, arrive very readily at the general term upon the prin- 

 ciples above explained^ 



Thus, 



