1 60 Algebra. 



Hence ;= -^~ ^ + i + .r + x" + x" + • . . 



x—x 



and we would obtain this very result if should assume 



or in other words if we begin the assumed series with a term con- 

 taining x'^ instead of beginning w4th an absolute term. If the 

 fraction we wish to develope is in its lowest terms, and if the 

 lowest power of x that appears in the denominator is the rth 

 power then we must begin our assumed series with a term con- 

 taining x~''. 



This is a safe rule whether the fraction is in its lowest terms or 

 not, but it i's not always necessaiy when the fraction is not in its 

 lowest terms. 



In any case when we form an equation by putting a given frac- 

 tion on the left and an assumed series on the right side of the sign 

 of equality, the assumed series must begin with such a power of 

 X that when the equation is integralized the lowest power of x on 

 the right side of the equation will be as low as the lowest power 

 on the left side. 



II. Examples. 



i-{-x-{-x^-^x^ 

 I. Develope 



Develope ^^, , ^^ 



X^-{-2X'^ 



x-\-zx' 



X'-\- 2X'' 



12. Not only fractions but some irrational expressions may be 

 developed by the method of undetermined coefficients. 



Let us develope \/ i — x. 

 Assume 



\/T^x=A^-^Ax+Ax' + Ax' + Ax'-i-A_x'=+ . . . 

 Square each side and we get 



i-x=A-f2AA-^+(2AA+A;K+(2A>3+2AA>^ 



+ (2A A^ + 2A A3-f A;>-^4-(2A„A^ + 2A A^-f2A^A3).r5+ . . 



Equating coefficients of like powers of x we get 



