Undktermixed Coefficients. 157 



is usually referred to as the method of imdetei^mined coefficients. 

 We will illustrate the method by working an example. 



lyet us develope the fraction 



I —X 



Assume 



j_^^=A„ + AA-+AX-f . . . +A_iJt-"-i+Rji-". (1) 



Multiply both sides oi (\) by \—x and we get 

 i=A +rA,-AJ.r + rA,-A,>-^+ . . . 



+ rA_i-A„_2>"-i+rR-A.,_i>-"-RA-"-^ 

 We see that the left hand member contains no power of x ex- 

 cept the zero power, or, in other words, in the left hand m^iber, 

 the coefficients of the various powers of x except the zero power 

 are each zero. Hence equating coefficients we get 

 A=i 



A-A=o.-. A=A, 



A-A=o .'. A=A^, 



A-A,=o.-. A3=A^', 



etc., etc. 



From these equations the law of the series is so evident that 



we can write as many more equations as we please without further 



calculation. 



We thus see from the second column of equations that each 

 coefficient equals the preceding one, and as the coefficient of .v', 

 or the absolute term, equals i; therefore each of the other coeffi- 

 cients equals i . Hence w^e obtain 



\—x 

 As we usually determine only a few of the coefficients, and then 

 discover if we can the law' of the series, so it is usual in the 

 assumed series with undetermined coefficients to write only a few 

 terms and indicate the others including the remainder by dots 

 thus: 



-- =A,.4-A A-+A A-^+A a-^+ . . . 

 \—x . - ^ 



Instead of using the method of undetermined coefficients we 

 might have proceeded by ordinar>' long division as follows: 



