206 
POWER SERIES 
, U 1 . U 8 . UÏ . u 5 . 
æ = “ + 2T + sT + r! + 5l + 
l+*= e > = l + ii+l 1 + - 
which agrees with 3). 
Thus we get 
x = 
But from 1) we have 
(3 
Taylor s Development 
169. 1. We have seen, I, 409, that if f (pc) together with its 
first n derivatives are continuous in 51 = (« < b), then 
/0 + 4)=/(«)+ ^/O) + |V"(<0+ - + J ^r,r-'X<0 
n— 1 ! l 
+ —J w (.a + 0K) 
n I 
(1 
where 
a<a+h<b , 0 < 0 < 1. 
Consider the infinite power series in h. 
2’=/0) + ^/'0) + ^/"0)+ - 
(2 
We call it the Taylor's series belonging to fix'). The first n 
terms of 1) and 2) are the same. Let us set 
R. = —.f in 0 + 04). 
n ! 
(3 
We observe that R n is a function of w, /¿, a and an unknown 
variable 0 lying between 0 and 1. 
We have /(a + h} = T„ + R n . 
From this we conclude at once : 
If 1°, fix) and its derivatives of every order are continuous in 
5( = (a, b), and 2° 
lim R n = lim — f (n) ia + Oh) = 0 , n = oc, (4 
a < a + h <b 0 < 0 < 1.