Undetermined CoefficienTvS. 155 



two series approaeh the SAME limit as the number of terms increases 

 ivithout limit, then the coefficients of like powers of x in the two series 

 are equal each to each. 



Since we are dealing with the limit of convergent series as the 

 number of terms increases without limit, we know that by taking 

 a sufficient number of terms the sum of the terms taken may be 

 made to differ from the limit of the sum by an amount as small as 

 we please. 



Let us then write 

 limit (A^-f A,-v + A x=-f . . . +A„_ijr"-'+ . . . ) 



=A^,4-A,.r+Ax^+ . . . +A,,_i.Y"-'-hR,x", 

 where V^x" is of course the diiference between the limit oi the sum 

 as the number of terms increases without limit and the actual sum 

 of the first n terms. 



A^, Aj, . . . A„_i are each constant, but R^ is not constant, 

 for if it were the series would terminate. In fact ^x" approaches 

 zero as n increases, for if it did not the series would not be 

 convergent. An inspection of the series shows that every term 

 after the first contains x, every term after the second contains x-, 

 ever>' term after the third contains x^, and so on ; hence every 

 term after the n th will contain the factor x'\ and so it is natural 

 to assume the remainder after 71 terms are written to be of the 

 form R^jf". 



Instead of writing limit of A^ + A^.r-f- ... as the number of 

 terms increases without limit we write 



A„+A x+A x=4- . . . A„_i.t"-' + R,-r" 

 and in the same way write 



B^.+BA-+BA--h . . . +B„_i.r"-' + R,^r" 

 instead of writing limit of B^ + B^.v + Bx'' . . .as the number of 

 terms increases without limit. 



Using this notation we may write 

 A^,4-A A-+ . . . H-A„_iA-"-' + RX' 



= B„H-Bx+ . . . +B„_i.v"-' + R^.r". (i) 



If now we consider .v as a variable approaching zero we have 

 here two variables which are always equal, and therefore by 

 Chapter IX, Art. 7, their limits are equal. By Chapter IX, 

 Art. 26, the limit of left-hand member equals A^, and the limit 

 of the right-hand member equals B_^* hence A^,= B^. 



