192 Algebra. 



By Art. 16 these two expressions must be equal, therefore 

 equating coefficients of like powers in (2) and (3) we get 



etc. 



NowMf we make h—o in (i) it is easy to see that A—/(x), 

 where yj'.vj means the same function of x that the given function 

 is of Ji:+^, or in other words, /(x) is what the given function be- 

 comes when k is put equal to zero. 



If, in (i), we substitute /('-X'j for A., and for the coefficients of 

 the various powers of// the values found in equations (4) to (8), 

 we get 



This result is Taylor's formula and is often spoken of as Tay- 

 lor's theorem. 



18. Application of Taylor's Formula. 

 Let us develope (x-^-k)^ by Taylor's formula. 

 Here /(x-^/i) = (x-^/i)\ 



Therefore f(x)=x\ 



Finding the successive derivatives of x^ we get 



Bj(x) = 6x\ T)lf(x) == 6 . 5X\ 



Dy^r; = 6.5.4-r^ Dy^r; = 6.5.4.3A-, 



B^./(x) = 6.s-4-3-2x, D«/r.^-; = 6.5.4.3.2.I, 



and every derivative after the seventh will equal zero. Therefore 

 by substitution in Taylor's formula we get 



(x + /i)'=x' + 6x^/1 + ^ ' ^x'/r 4- -^* ^-trVz^ 



2.3.4 2.3.4.5 2.3.4.5.6 



or (x-\-/f/=x^-i-6x~ii+isx'/r+ 20xVi'-\- 1 5x^/1' -\-6x/r-\-x\ 



