222 
POWER SERIES 
To prove 6) we have from T) 
/ (n) (a + a)=/ (n) (a) + oc/* (w+1) (a) + |r/ (n+2) 0) + ••• 
and from 4) 
Gr (n w (a) = |/ (n) (a) I + « 
y(ra+l)( a ) 
+ fyj/ (n+2) («) 
The comparison of 7), 8) proves 6). 
+ 
a 
(8 
Circular and Hyperbolic Functions 
183. 1. We have defined the circular functions as the length 
of certain lines; from this definition their elementary properties 
may be deduced as is shown in trigonometry. 
From this geometric definition we have obtained an arithmeti 
cal expression for these functions. In particular 
x x 3 , x b x 1 , 
sin x = b 
1 ! 3 ! 5 ! 7 ! 
1 x 2 . x* X 6 
+ 
(I 
(2 
valid for every x. 
As an interesting and instructive exercise in the use of series 
we propose now to develop some of the properties of these func 
tions purely from their definition as infinite series. Let us call 
these series respectively iS and C. 
Let us also define tan x — seca; = —-—, etc. 
cos x cos x 
2. To begin, we observe that both S and C converge absolutely 
for every x, as we have seen. They therefore define continuous 
one-valued functions for every x. Let us designate them by the 
usual symbols 8ina: , cos:t . 
We could just as well denote them by any other symbols, as 
<K*) ’ ^0*0- 
3. Since s=0 , C=1 for a; = 0, 
sin 0 = 0 , cos 0 = 1. ' 
we have