220 
POWER SERIES 
182. Let /(x) have finite derivatives of every order in 
2{ = (a<6). In order that/(a;) can be developed in the Taylor’s 
series £ 2 
/(*)=/(« +A) =/(«) +A/'00 + jjrj /"(«) + ••• 
valid in the interval 21 we saw that it is necessary and sufficient 
that lim R n = 0. 
But R n is not only a function of the independent variable A, but 
of the unknown variable 6 which lies within the interval (0, 1) 
and is a function of n and A. 
Pringsheim has shown how the above condition may be replaced 
by the following one in which 6 is an independent variable. 
For the relation 1) to be valid for all h such that 0 it is 
necessary and sufficient that Cauchy’’s form of the remainder 
R.(h, $■) = (1 ~ + eh-), 
the h and 0 being independent variables, converge uniformly to zero 
for the rectangle D whose points (A, d) satisfy 
0 <h<H 
O<0<1. 
1° It is sufficient. For then there exists for each e>0 an m 
such that 
| 0) | <e n^>m 
for every point (A, d) of D. 
Let us fix A ; then | R n | < e no matter how 6 varies with n. 
2° It is necessary. Let A 0 be an arbitrary but fixed number in 
21 = (0, if*). 
We have only to show that, from the existence of 1), for h<_h 0 , 
it follows that 
R n (h, 6) = 0 
uniformly in the rectangle P, defined by