TAYLOR’S DEVELOPMENT 
207 
Then 
/(a + A)=/(a) + A/(a) + |!/"(a)+ ... (5 
The above theorem is called Taylor s theorem; and the equa 
tion 5) is the development of f(x) in the interval 51 by Taylor's 
series. 
Another form of 5) is 
/(*)=/(«)+i^ rT ^/(«)+^ a i 2 /''(«)+... (6 
When the point a is the origin, that is, when a = 0, 5) or 6) 
gives 
/(*) =/(«) + */'(«) +fy"(0) + - (7 
This is called Maelaurin s development and the right side of 7) * 
Maclaurin's series. It is of course only a special case of Taylor’s 
development. 
2. Let us note the content of Taylor’s Theorem. It says : 
If 1° f(x) can be developed in this form in the interval 
2l = (a<5); 
2° if f(x) and all its derivatives are known at the point 
then the value of / and all its derivatives are known at every 
point x within 51. 
The remarkable feature about this result is that the 2° condi 
tion requires a knowledge of the values of f(x) in an interval 
(a, a + 8) as small as we please. Since the values that a func 
tion of a real variable takes on in a part of its interval as (a < c), 
have no effect on the values that/(2;) may have in the rest of the 
interval (c < b), the condition 1° must impose a condition on f(x) 
which obtains throughout the whole interval 5b 
170. Let f(x~) be developable in a power series about the point a, 
viz. let 
/(x) = a 0 + a x (x — a) + afx — af + ••• 
(1 
(2 
i.e. the above series is Taylor's series.