130 ALC^EBRA. 



c. Reduce • to an infinite series. 



Ans. I-|•2A;-{-2^^+2^^-J-^^^ ^c. 



a 



6, Reduce — ; — ri to an infinite series* 



Ans. I Y ~ —-, ^ 





2« * A?^ 



7. Reduce j to an infinite series* 



PROBLEM 



■» — T> or -J ; two terms are deficient by -r^-2 ; three terms exceed 

 it by tV ; four terms are deficient by ^^ > five terms will exceed 

 the truth by -^-V* ^c. So that each succeeding term of the serieiJ 

 brings the quotient continually nearer and ftearer to the truth bjr 

 ttit half of its last preceding difference ; and consequently the 

 series will approximate to the truth nearer than any assigned num- 

 ber or quantity whatever ; and it will converge so raiich the swift- 

 er, as the divisor is greater than the dividend. 



But the second series perpetually diverges from the truth ; for 

 the first term of the quotient exceeds the truth by i — f, or 4 » 

 two terms thereof are deficient by f ; three terms exceed it by 4 » 

 four terms are deficient by ^-/ ; five terms ejCceed the truth by 

 V, &c. which shew the absurdity of this series. For the sanae 

 ncason, x must be less than unity in the second example ; if at 

 were there equal to unity, th^n the quotient would be alternately 

 I, and nothing, instead of \ ; and it is evident, that x is less than 

 \inity in the first example, otherwise the quotient would not have 

 been afnimative ; for if ,v be greater than unity, then i — x, the 

 divisor, is negative, and unlike signs in division give negative quo- 

 tients. Tiom the whole of which it appears, that tlie greatest 

 term of the divisor must always stand first. 



