226 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



suit which it is concluded would follow, in like manner, for all the other coeffi- 

 cients. 



But here it will be at once seen that the expansion (b) which is unexception- 

 able whilst y is small, cannot, except on Mr. Peacock's principles, be assumed 

 as holding good where j is infinite; and that, even assuming this expansion, it 

 may happen that a is infinite, a result which would reduce an infinity of the 

 first terms of (b) to zero, and thus cause the series to have both numerators and 

 denominators infinite, and consequently render it capable of a finite sum. That 

 such is really the case may be readily shown, for, writing 



and putting 

 we have 



and consequently 



e^^ = (e^) 



e' = X 



e^ = x '°^: 



-1 



xa 



= /— ±\ X' (— 2 log X -1- log — 1) 

 = + L 



X x^ (log X — ^ log — I) 



or 



1 



x^ 



= + 



■y x^ log X 



x = 



but putting X = - we have 



1 y 



z 



x'' log X log y 



and as y^ when y is infinite, is infinitely greater than log y, we have 



x = 



whence we conclude that if the function in question is expanded in a series of 

 the kind proposed, the lowest power of x must be infinite. And as log x is, in 

 like manner, a function that expands by infinitely small powers of x, it would 

 appear that we might take, as the classifying functions whereby developments 

 can be arranged, the simple expressions 



+ \ 

 x % e ^", (log x)°. 



