Infinite Series . 8 5 
may be purfued, when the denominator of a fluxion, which is 
a fun&ion of x multiplied into x contains the Ample power 
only of a fa&or or fa&ors ; reduce the factor or fadlors into an 
infinite feries, proceeding according to the dimenfions of x 9 and 
by the known methods find the fluent of the fluxion* 
1 1. The fluent of the fluxion or fum of the feries may be de* 
duced alfo from the fubfequent propofitions, from which may 
be inveftigated many feriefes, whofe fums are known, 
1 • Let pP — Qq, qQ = Rr, rR = Sf, sS = T/, &c ; then f P p » 
Pp - Qq +- Rr - Sj + T/ - &c. if only the feries converges. 
2 . Let pP'-\-p / P / = Qq / , qQ + q'Q + = Rr', rR' + r'R' — 
S /, jS / + /S / =T/ / , &c. where P', />', Q., q\ Sec. denote the 
increments of the quantities P, p 9 Q*, q, See. refpe&ively, then 
will the integral of the increment (P/ ) = Pp - Qq + Rr - Ss 4- 
&c. if only the feries converges. 
Ex. 1, fx-’x = f±L = J- (-±- + A + i±^R+ 
I ^ x n ~~ l V 7/2+1 r + I j + I * + 1 
C+^D + &c.) ==^-; whence — h- = — — I- n -^ X — — 
T+i J 1 __ 1 1 — n m + 1 r + 1 m I 
— h&c. In this example i = P ~x~~ n ~ m 9 
s' + 1 r + 1 m 4 - 1 * r 9 
• ® 
q=x r 'x, Q= x~ n ~ r ', r — x*'x, &c. 
Ex. 2. / fJL = f ^~ n ~ r x- 1 : 
^ «+*■ ’ m — n — r+ 1 
8— >*+i 
Xm-b 1 722+1 
?2 + r n-\rr- j- 1 
+ &c.) 1 whence 
7724-2 ‘ 7/2+ 1 m+2 772+3 
. L_. — = — i_ 4. J± r A + — Ltifi + &c. In both thefe exam- 
m — n—r^r i m + j 772 + 2 772 + 3 
pies the letters A, B, C, &c, denote the preceding terms. 
•c>„ „ /V* *’” +I / j , 1 * 2 
£<X« 2» / ss — " I — — — 4* ■ 1 ■ "* • " x ” + “". irT— ^ V 
•>,/ 1+A- 1+^ 772 +1 ^+ 1 . 772 +2 1 + * 772 + I .02+2, ^+3. 
0.0 v 3 
2 ■ 3 
(1+#) w + i .772 + 2,772+3.772 + 4 (1+*) 
. + &c.) 
(I+*) s * 
Ex, 
