158 Algebra. 



.r I i-f.r + .v--f.t- + .r^ 



I— a- 



X 



X — x'- 

 X- 



x' — -V^ 



.r^ — -f* 

 x\ 



Here as before we obtain 



I— ,1" 



This series on the right side of the sign of equality is conver- 

 gent if a'<i, but not otherwise, and therefore this series cannot 



be called the development of unless .v is less than unitv. 



I— A- 



See Art, 7. When x is equal to or greater than unity the fraction 

 cannot be developed. 



9. Examples. Develope the following fractions both by the 

 method of undetermined coefficients and by actual division, and 

 in each case discover the law^ of the series. 



Also in each case state for what values of .v the series is a true 



development. 



I i+o: 



I. — . 6. . 



i+jt- I— .r + .i- 



i+.v 1+ 2a- +3.1"^ 



\—x' \—2X-\-2)X^' 



a \ — 2X^r^x^ . 



^' a-x' ■ ' i + 2.i-f3-r^" 



2 + 3.1- 5.1-4-7 * 



'^' i+.r' 3 — 4x''4-2.r 



I 8x~ — -xx^ 



5. . ... , . ... ^o. ^ 



I — 2 A -h sx-" 3 — 4.x^ + 2X-' 



Compare the laws of the series in the developments of the frac- 

 tions in examples i and 4 ; also compare the laws of the series 

 in examples 5 and 7 ; also in examples 9 and 10. 



