82 CAUCHY'S PROOF OF TAYLOR'S THEORFJU. 



ble, and that all the differential coefficients up to the 71 th in- 

 clusive vanish when x = a ; we have by Art. 106, 



Suppose a = and F(a) =0, then 



108. Application io Taylors Theorem. 



Let < (x + h) be a function which is to be expanded in 

 a series of ascending positive integral powers of h. Let 



Then F(k} and its differential coefficients with respect to h, 

 up to the n th inclusive, vanish when h = 0. Also 



Hence, by the last equation of Art. 107, 



and therefore 



From this Taylor's Theorem follows whenever the func- 

 tion is such that, by sufficiently increasing n, the term 



can be made as small as we please. 



