70 TAYLOR'S THEOREM. 



Hence, substituting the values ofA <) ,A l ,... in (l),wc have 

 f(x + h) =/(*) + If (x) + ^f" (x) + Jj^f" (*) + ... (4), 

 the general term being 



[n dx n 

 This result is called Taylor's Theorem. 



88. There are numerous objections to the method of the 

 preceding Articles, and especially the use of an infinite series, 

 without ascertaining that it is convergent, is inadmissible; \ve 

 proceed then to a rigorous investigation. 



89. Let y = F (x), and suppose Ace and Ay to represent 

 the simultaneous increments of x and y; then the fraction 



Aw 



j*- , since it has for its limit the differential coefficient F' (x), 



will ultimately when A# is taken small enough have the same 

 sign as this limit, and therefore will be positive if the dif- 

 ferential coefficient be positive, and negative if the differential 

 coefficient be negative. In the former case, the quantities 

 Ay and Ax being of the same sign, the function y will increase 

 or diminish according as x increases or diminishes. In the 

 latter case, Ay and Ace being of contrary signs, y will increase 

 if x diminishes and will diminish if x increases. 



The above supposes that there really is a finite limit to 



Ay 

 which -r* tends ; in other words we assume that F' (x] is not 



Ax 



infinite. The limitation that the functions with which we are 

 concerned are not to become infinite is one which ought to be 

 understood in most theorems in mathematics, even if it is not 

 formally enunciated. , In the present subject however it is 

 usual to state this limitation expressly at the more important 

 stages of the investigations. 



It may be observed that we may sometimes obtain useful 

 information respecting the sign of a function by examining 

 the differential coefficient of the function. For example, 



suppose y = (x 1) e* + 1, then -Jf- = xe" ; as ~ is positive 



7 */f * C*O 7 



ax ax 





