FAILURE OF TAYLOR'S THEOREM. 85 



If then a, /9, 7, be all positive integers, we shall never 



have negative powers of k introduced by successive differen- 

 tiation of f(a + h). Hence, by putting h = 0, we introduce 

 no infinite values. 



But if any one of the exponents ct, /3, 7, ... be negative, 

 f(a + h) and all its differential coefficients contain negative 

 powers of h, and therefore/(a), f (a), f" (a), ... are all infinite. 



If none of the exponents be negative, but one or more of 

 them be positive fractions, suppose that 7 is the smallest of 

 such fractions, and that it lies between the integers n and 

 n + 1. Then f (a + h) and all its differential coefficients up to 

 the ?i th inclusive are free from negative powers of h; but 

 f" +1 (a + h) and all the subsequent differential coefficients con- 

 tain negative powers of h. Hence f n+1 (a) is the first differen- 

 tial coefficient that becomes infinite, and all the following 

 differential coefficients are infinite. 



113. It will be of use hereafter to remark that if for a finite 

 value of the variable any function becomes infinite, so also 

 does the differential coefficient of the function. In proof of 

 this, it is sufficient to notice the different cases that may arise. 

 An Algebraical function can only become infinite, for a finite 

 value of the variable, by having the form of a fraction the 

 denominator of which vanishes. Now when we differentiate 

 a fraction we never remove th6 denominator, so that the 

 differential coefficient also has a vanishing denominator, and 

 therefore becomes infinite. Similarly, the second, third,, . . . 



differential coefficients are also infinite. 



i 



The transcendental functions logo; and a x , which both 

 become infinite when x=0, have their differential coefficients, 



namely - and ~- a*, also infinite when x = 0. 



J x a? 



The trigonometrical functions, such as tana; and sec a?, 

 which can become infinite, are fractional forms, and fall under 

 the observations already made. 



The proposition is not necessarily true for functions which 

 become infinite for an infinite value of the variable, as may be 

 seen in the case of log x, which is infinite when x is infinite, 



while its differential coefficient - vanishes. 



x 



