Page:The method of fluxions and infinite series.djvu/27

and INFINITE SERIES, 3 . ..ift Av Examples of Reduttion by Dhifwn. IjfM/l^^ '* / .4. The Fraction ^ being propofed, divide aa by b + x in the following manner : faa aax aax 1 a a x* aax* . " . aax aax O --7 -f-O aax* o -+- o - +o flt * ** Jf* ~ ;. -rr^i_ *- " v i r * ^^ tf*^ 1 a* x* a* x* . a* X+ ~ The Quotient therefore is T _- JT - + - T _ . rr + T7 -, &c. which Series, being infinitely continued, will be equivalent to j^. Or making x the firft Term of the Divifor, in this manner, x + toaa + o (the Quotient will be - - ?4 4. 1^ V & c ~ e , , % r~ _ _ * ** n * AV found as by the foregoing Procefs. 5. In like manner the Fraction ~- will be reduced to I # -{- x 4 ' A:* H- x 8 , &c. or to x-* #-* _f. ^- ^-8 2 * " 9 v 6. And the Fraction r will be reduced to 2x^ 2x i s i+x* 3* + yx 1 13** -j- 34x T , &c. 7. Here it will be proper to obferve, that I make ufe of x-', x-', x-', x-*, &c. for i, ;r 7,' - &c. of xs, xi, x^, xl, A 4, &c. for v/x, v/*S / x *> vx , ^x l , &c. and of x'^, x-f. x - i &c for , i j_^ ' * **** 1Ui ^ x ^ ? >' y-^.' &c. And this by the Rule of Analogy, as may be apprehended from fuch Geometrical Progreflions as thefe ; x, x*, x> (or i,) a"*,*-',*'*, *, &c. B 2 8. x,