200 
POWER SERIES 
Hence 
1 — («o + a 2^ 2 + ‘ ’ O ( 1 — + ^7 
X 2 
= a 0 + (“» - fj/ + (“4 - ft + ft/ 
+ ( a ‘-f! + fl-7!y + (“*-^ + fl-n + fiy + 
Comparing like coefficients gives 
ük — o 
3! 
1 
a 2 = 7i- 
b 
360' 
Thus 
a _ /i + _ a 2 _ Za = 0 .-.a = ^ 
6 3 ! + 5! 7! ' 6 3-7!' 
1 = 1 _i_ 1 ^ _i_ _!L ^ i _§J_ ^5 I ... 
sin x x 6 360 3 • 7 ! 
valid in ( — 7T*, 7r*). 
166. Let 
where 
(4 
^C*0 =/!(>)+/2 <>)+ 
/»0*0 = «„o + a Bl * + a n2 x 2 + ••• n = 1, 2 ••• 
Let the adjoint series 
«»o 4- a ni£ + <*»2! 2 + * * • 
converge for % — R and have </> n as sums for this value of f. 
<& = (f) 1 + (f> 2 + ••• 
converge. Then J 7 converges uniformly in 51 = (— R, R) and F 
may be developed as a power series, valid in 51, by summing by 
columns the double series 
a 10 + «11* + «la* 2 + ”• 
4” **20 4” **21*^ 4" **22^ 2 4~ ■■■ 
+ a 30 + a 31 x + a 32 x 2 + ••• 
(1