58 DIFFERENTIAL AND INTEGRAL CALCULUS. 



tions of x only ; then differentiating both sides of this 

 equation (1) with respect to h only, not assuming x to 

 vary (2) with respect to x only, we shall have 



(1) 



dx dx \n- 



Since the differential coefficient of < (z) with respect to h 

 is <' (z) -77 , and with respect to x is </>' (z) -y- , and when 



T dz ^ dz 



z = x + h, -=- 1, and -77 = 1. 

 7 a j; aA 



From these two expansions of the same function we get 

 dA^ dA. d*A^ 



Also putting ^ = in the original expansion, we get 

 A Q = (j)(x}j or the expansion, if such a one can be effected 

 at all, is 



No information is given by this method as to when this 

 series is divergent, and when convergent, and arithme- 

 tically true, and so long as no satisfactory interpretation 

 is given to divergent series, we cannot assert the equality 

 of cf)(x + h) to the series which professes to be its ex- 

 pansion in powers of ^, until we have ascertained that 

 the series is convergent, and tends to a finite limit what- 

 ever the number of terms taken. Thus if we expand 



by this series, we get of course the expansion 

 1 h tf h* 



