154 Al^GEBRA. 



Again let x approach zero as a limit, then 



Hmit of A, + A,-r-|- . . . +A„a"-'=A„ 

 limit of B^ + B^x+ . . . +B„.r"-' = B,, 

 therefore A^=B, by Chap. XI, Art. 7. (s) 



Subtracting A^ from the left side and B^ from the right side of 

 (4) we get 



Ax-\-Ax'-{- . . . -i-A„x"-'=^Bx-j-'Bx'-\- . . . -\-B,a-"~' (6) 

 Divide (6) by x and we have 



A +A3Jt-+ . . . +A„.r"-^=E^ + Bx+ . . . -f-B„A-"-^ (j) 

 Then in same way as in the two preceding instances it follows 

 that A^=B^ 



and bv continuing the process we get 



A=B, 



etc. 

 Therefore if the two functions are equal for all values of x, the 

 coefficients of like powers of x in the two functions are equal 

 each to each. 



3. Equations of the kind just considered, which are satisfied 

 by a?ij/ value of x are often called Identical equatioyis, while those 

 with which algebra has most to do, those satisfied by particular 

 values of x equal in number to the degree of the equation, are 

 often called Conditiorial equations, 



4. Definitions. A Series is a succession of terms each of 

 which is derived from one or more of the preceding ones by a 

 fixed law. An Infijiite Series is one in which the number of terms 

 is unlimited. 



An infinite series is Co?ivergent if the sum of the first n terms 

 approaches a limit as n increases without limit. 



An infinite series is Divergent if the sum of the first ;/ terms 

 does not approach a limit as 71 increases without limit. 



The series \ -\- x -\- x^ -\- . . . is convergent if x is less than unity, 

 but divergent if .v is equal to or greater than unity. 



5. Theorem. If for every value of x ichicli makes each of the 

 two series A -\-A x-\- . . . and B -\-B x-\- . . . convergent these 



