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CHAPTER VI. 

 "EXPANSION OF FUNCTIONS IN SERIES. 



84. IN the Binomial Theorem, we are furnished with a 

 series proceeding according to powers of h, which is equi- 

 valent to the expression (x + h} n . Other series have also 

 presented themselves in Algebra and Trigonometry, such as 

 the expansion of e x in powers of x and of log (1 + x] in powers 

 of x. In the previous Articles of this book, we have, however, 

 not assumed the knowledge of any expansions, except the Bi- 

 nomial Theorem in the case of a positive integral exponent ; but 

 we are now about to investigate the expansion of f(x + h) 

 in powers of h, where f(x) denotes any function of x, and it 

 will appear that all the isolated examples which the student 

 may have seen hitherto, are but cases of this general theorem. 



85. Before we offer a strict demonstration of the theorem 

 in question, we shall notice the method which it was usual to 

 adopt in treatises on the Differential Calculus not based on 

 the doctrine of limits. Such treatises commenced with an 

 unsatisfactory demonstration of the proposition that f(x + A) 

 could generally be expanded in a series proceeding according 

 to ascending integral positive powers of h ; it remained then 

 to determine the coefficients of the different powers of h, and 

 that was accomplished in the manner given in the next two 

 Articles. 



86. We have first to establish the following theorem. 



h) be any function of x + h, we obtain the same 

 result whether we differentiate it with respect to x, consider- 

 ing h constant, or differentiate it with respect to h, consider- 

 ing x constant. 



