INFINITE SERIES. 3^9 





4. Reduce — to an infinite series. 

 ~ a — ;v 



5. Reduce 



, I ■ ■ • ' ■ ■ - ■ ... ■ ■- 



term of the quotient, by which we also maltiply a-{-x, the divi- 



aix cx^ ex cx^ 

 sor, and the product is — — r 7-, or — ■ , , 



which, being taken from , leaves H . 



a a* 



The rest of the quotient is found in the same manner ; and 

 four terms being obtained, as above, the law of continuation be- 

 comes obvious 5 but a few of the first terms of the series are gen- 

 erally near enough the truth for most purposes. 



And in order to have a true series, the greatest term of the di- 

 visor, and of the dividend, if it consist of more than one term, 

 must always stand first. 



Thus in the last example ; if a: be greater than ^, then x must 



be the first term of the divisor, and the quotient will be zz: 



*+^ ' 

 € ac , a^c ^^ . « t . f -r , , 



— r H J 37- 4-,&c. the true series : but \ix be \tyi 



X X- x^ x^ 



than <z, then this series is false, and the further it is continued, 

 the more it will diverge from the truth. 



For let tf=:2, cr=L\ and xzz\ ; then if the division be perform- 

 ed with a, as the first term of the divisor, you will have -*- — •i::^ 



(, a-^-x 



c 



X'\-a 



But if X be placed first in the divisor, then will ■ - 'j - =: §• 



— Y- =1— 24-4 — 84-16—, &c. 



l4-2 



Now it is obvious, that the first series continually converges \q 

 the truth ; for the first term thereof, viz. t, exceeds the truth by 



S s 



