TAYLOR’S DEVELOPMENT 
219 
Since Taylor’s series T is a power series, it converges not only 
in 21, but also within 58 = (2 a — b, a). It is commonly supposed 
that = T also in 58. A moment’s reflection shows such an 
assumption is unjustified without further conditions on f(x). 
2. Example. We construct a function by the method considered 
in I, 333, viz. 
(1 
Then /(») = cos x, in 21 = (0, 1) 
= 1 4- sin x, within 58 = (0, — 1). 
We have therefore as a development in Taylor’s series valid 
* n f( x ~)z=l— ——= T 
/( - ' 2! 4! 6! + 
It is obviously not valid within 58, although ^converges in 58. 
3. We have given in 1) an arithmetical expression for f (x). 
Our example would have been just as conclusive if we had said : 
Let /(#) = cos x in 21, 
and = 1 + sin x within 58- 
181. 1. Criticism 5. The following error is sometimes made. 
Suppose Taylor’s development 
/00-/00+oo+ O) + - (i 
valid in 21 = (a < b). 
It may happen that T is convergent in a larger interval 
58 = («<£). 
One must not therefore suppose that 1) is also valid in 58. 
2. Example. 
Let 
f(x) = e* in 21 = (a, b), 
= e x + sin (x — 5) in 58 = (&, -B). 
and 
Then Taylor’s development 
rJl /y*3 
■/<»)-i+fi + ff + f7+- 0 
is valid for 21. The series T converging for every x converges in 
58 but 1) is not valid for 58-