84 FAILURE OF TAYLOR'S THEOREM. 



of the variable lying between certain values, the demonstration 

 given of the theorem 



is no longer valid. It is usual to speak of the cases where an 

 infinite value enters as " instances of the failure of Taylor's 

 Theorem." The phrase is connected with the imperfect mode 

 of demonstration given in Arts. 86 and 87, in which it was 

 not settled beforehand when the theorem supposed to be 

 demonstrated was really true and when it was not. For ex- 

 ample, suppose 



so that f(x + h) = \/(x a + h). 



Then it would be said thaty(# + K) can always be expanded 

 in a series of whole positive powers of h, except when x = a. 



"When x = a, f (x), f" (x), ... all become infinite, and 

 f(x + h) becomes ^h. 



112. It was usual in that system of treating the Differen- 

 tial Calculus referred to in Art. 85, to express, or imply, 

 two propositions with respect to the "failure of Taylor's 

 Theorem." 



(1) If the true expansion of f(a + h) in powers of h 

 contain only integral positive powers of h, then none of the 

 quantities /(a),/' (a),f"(a), ... can be infinite. 



(2) If the true expansion of f(a + h) in powers of h 

 involve negative or fractional powers of h, then some one of 

 the quantities f(a), f (a), /" (a), ... is infinite, as well as 

 all which succeed it. 



By the true expansion of f (a + h) is meant the expansion 

 obtained by some legitimate algebraical process, applicable to 

 the example in question, as the Binomial Theorem for example. 

 The proof of the above two propositions was given thus. 



Suppose f (a + K) = A + AJi a + AJi? + AJif + 



to be. the true expansion, A , A lt ..., not containing h. Then 

 to obtain f (a), f" (a), ... we may differentiate /(a + A) 

 -successively with respect to h, and put h = in the result 



