198 
POWER SERIES 
if fix) = % + a \ x + «2^ +•••; — R <x < R, 
(1 
is an even function, the right-hand side can contain only even powers 
of x; if f{x) is odd, only odd powers occur on the right. 
For if/ is even, 
But 
fix') =fi~x). 
fi — x) — a 0 — ape + a 2 x 2 — 
Subtracting 3) from 1), we have by 2) 
0 = 2 iape + apt? + aph + • • •) 
for all x near the origin. Hence by 160, 2 
= a 3 = a h = ••• =0. 
The second part of the theorem is similarly proved. 
165. Example 1. 
Since 
fix) = tan x. 
sin x 
tan x = 
cos x 
and 
we have 
tan x = 
x ah , ah 
sinz=^-^+~-- 
. X 2 x i 
oos * =1 -2! + 4! - 
a? , ah 
*~3! + 5!~ 
x 2 ah _ 
“2T4! " 
1+ Q 
(2 
(3 
(1 
Since cosz>0 in any interval 33 = - + S, S>0, it 
follows that 
in 33. 
K?I<1 
Thus by 163, 2, tan x can be developed in a power series about 
the origin valid in 33- We thus set 
tan x = ape + apth + apah + • • • 
(2