RECURRING SERIES. 



Tliu'i, -x^. an exnmple, let there be propofed the recurring wliitli tliereforc arifes from the fraftion 



ferics 



I + 4.r + 14. v' + 46 .v' + r46K* + &c. 



1 — Z X + x^ 



to find the general term, or the co-efficient of .r". C, a. t^ ■»•" _ C2 k — r 1 a-" + ' 



Tiie fiini of this feries is found by the preceding part of ' L^+ " '^ >^ " " •* 



this article to be 



(I -x)' 



I — 5 X + 6x^ whence the fum of n terms is 



J 2 



This rational fraftion is equal to + — —• Now , 4. .^ _ (2 « f 1) :>;" 4- (2 n ~ i)x" + ' 



, 2 c-^r 



the general term of ^-— ^ is - 2"^' ; and of ^—-j^ ^_ Required the fu.m of « terms of the feries 



is - 2 . 3".t% as fhewn above, therefore the general term , _ ^. j. /„ _ jX ^> , („ _ ,\ j,.i ^ &c. 



of the propofed feries will be (2 . 3" - 2") .r': ^ / • V / ^ \ 3/ 



It may be obfervcd, however, that this method is in fome J^ere again the fcalc of relation is + 2 — i, tiierefore the 



cafes very laborious, and not always prafticable ; it will, infinite Turn is 



therefore, frequently be better to make ufe of the multinomial . _ , W. 1 f „ _ 2) a;' — 2 (n — I ) a-' 



theorem, wliich may be done by putting the generating frac- L — -J — 1 '"_ \i~ " — = 



tion under the form ^ ' 



{a + bx + cx' ^ &c.) (i -o.x-(ix'--yx'- &c.r' ( n - i).t-n.v ' 



See Multinomial Tuv.owvAU ^ ' 



IW.—Tofnd the fum of any number [n) terms of a recur- ^^^^^ ^ terms, it becomes - «" + ' - 2 y^""- 3cc. thefum 



ring ferics. ,, + 1 



For this purpofe it is only neceifary to find the co-efficient ^f ^^.j^jj-ji \^ found in the fame manner to be -j there- 

 of the n 4- ithtermof the feries. Then, from the fum of (1 — *)' 

 the entire feries fubtraft the fum of the ferics beyond the fore n terms of the propofed feries is 

 «th term, and the difference will obviouflybe the fum of the , > „ ,.1 1 ^« + « 

 rlt n terms lought. r^ 



Let there be propofed as an example to find the fum of (' ~ *) 



n terms of the feries Hence, alfo, n terms of the feries 



1 + 2.V 4- 3 ..' + 4-' + • • • • -"- • («-,)..(« - 2) .v^ ^ («ZLl)ji'4. w . 



The infinite fum of this feries is readily found = , _ - n n n 



^ "/ {n - 1) X ~ n x' ->r x'*' 



In the fecond cafe, viz. of the terms beyond n .v" ", the n[l — xY 



feries is ^ 



(« + 1 ) .v" f f n 4- 2) .r" + ' 4- (« + 3) x'^^, &c. ; ^n a fimilar manner the fum of « terms of the feries 



andintheformula. = " + ^ " " '^ ^; we have only to fub- 1 = + 2^r + 3'^^ + 4' *' + &c. 



' - '^ •* - ' * , ^ . ,. , . is found to be 

 ftitute a = (« 4- i) .V-, and (3 = (n 4- 2) .r'+S milead of 



a = I and /S = 2 X, as in the former. Hence we have i 4- x — (n + i)^ .r' -t- (2 n^ 4- 2 « — i) x"''' — «'«" + ' 



, (n 4- I) «•■ + (« +2).r»+--2(«+l).i^"^' (l - ^)^ 



S ^ — ■ — ~ : 'J ur 



I — 2 * 4- .*" the fcale of relation being 3 — 3, i. 



(n 4- i) a:" — nx"*' I" ^^ ^^^ preceding examples we have determined the in- 



-»' = — — / _ — 7y5 — ; finite fum of the feries, as beginning at the firft term, and at 



' , . the (« + I )th term, a more eafy method is as follows, which 



and, therefore, 1 — /, or the fum of the firft n terms is jg jue to Simpfon. Ellays, p. 96. 



equal to LetA+B4-C + D4-...+K4-L4-M4-Nbe 



1 _ (n 4. i) >:» 4. nx"'*'' any finite recurring feries, of which each term depends upon 



(j _ ^Y ' the three which precede it, the fcale of relation being^, q, r, 



fo that 



2. Required the fum of n terms in the feries 



/A4-?B4-'-C=D 



I + 3 K + 5 a' + 7 V + .... (2 « — I ) x"-'. pB + qC + rD = E 



Here, by trial, we find the fcale of relation to be ji = 



4- 2, and »z= — I, as before, therefore the infinite fum is ^^^ which is the fame, 



, _ »+ g -«/x« _ I + sx- zx _ I + -f ^ />A4-?B+rC-D = o 



I — fix — V x'' I — 2j;4-*'' (i— *)' /B4-5rC4-rD — E=o 



After n terms, the feries becomes 



/>C 4-jD4-rE-F=o 



(2b + l)a;" + (2 n + 3) .v» + ' 4- (2 n 4- 5) «"+*[- &c. pK.+ qL + rU— 1^=0 



whence. 



