74 MACLAURIN'S THEOREM. 



"We may, if we please, change h into x, and since the 

 quantities /(O), /'(O), ...... /"(0)> do not contains or h, no 



change is made in any of them : hence 



TV* (ti\ sr n+1 



/(*) =/(o) + xf (o) + ... t fflU2L+/- (fej. 



When the last term, by taking n large enough, can be 

 made as small as we please, we have forf(x) an infinite series 

 proceeding according to powers of x. This series is usually 

 called Maclaurin's, having been published by him in 1742; 

 though, as it had been given a few years previously by Stir- 

 ling, it sometimes bears the name of the latter. 



96. Assuming that any function of x can be expanded in 

 a series of positive integral powers of x, the following method 

 has been given for proving Maclaurin's Theorem. 



Let f(x)=A + A l x + A z x* + ...... + A n x n + ...... 



where A , A v ^^...do not contain x. 



Differentiate successively, then 



Now suppose x = Q in each of these equations, we have 



A = f" CO) 

 2 1.2* ( '' 



Substitute the values of A , A lt ... and we obtain 



