Series. 193 



This result is seen to be the same as that obtained by a direct 

 application of the Binomial formula, which of course is as it ought 

 to be. 



As a second example, let us develope \f x-\-h by Taylor's 

 formula. 



Here f(x-^h) = (x-^h)^ . 



Therefore f(x) =x'^. 



Finding the successive derivatives of x'-^ we get 



DJ(x)=ix ^ D^/(x) i.^x -^ 



etc. 



Making the substitutions in Taylor's formula we get 



J. _i _^ _A _j_ 



\/x-h/i=x--j-^x "h—\x Vz^+yV-^ '■^>^'''— yts-^-*^ '"//'+ • • . 

 If in this equation we make jr= i we get 



If in this equation we change the sign of h we get 



s/^-h^i-\h-\h^-i^l^—^i^h^- . . . 

 Compare this with the result obtained -in XII, Art. 12. 



It was stated in Art. 13 that Taylor's formula could be used to 

 develope any function of a binomial which is capable of being de- 

 veloped into a series arranged according to positive increasing 

 powers of one of the quantities. It is indeed a matter of substi- 

 tution, but care must be taken that the substitution be such that 

 the development obtained is arranged according to positive in- 

 creasing powers of the proper quantity. 



If, for example, we wish to develope V-^'+i into a series ar- 

 ranged according to positive increasing powers of x, it might at 

 first appear that, in the development of sf x-\-h, we could simply 

 make Ji=\ ; but this would give us a series arranged according 

 to positive increasing powers of i , not x. The proper course is as 

 follows : First, develope V x-\-h according to positive increasing 

 powers of h ; then in this result make a== i and we have the de- 

 velopment oi sf \-\-h arranged according to positiv^e increasing 

 powers of h ; then, finall}^ in this result, change // into .r and we 

 obtain the result sought. 



