sums of several classes of infinite series. 
275 
Nr n+I -{- &c. will have its terms each greater than 
the corresponding terms of the other series. Now this series 
is equal to 
/*f(^iog_L) [: = :] (F) 
I shall now apply some of the theorems to the investigation 
of the sums of series, and then explain a method which (when 
the equations to which it leads can be solved) will in all cases 
render them correct. And first let f( 0 ) = (cos 0 )”= (1- — ~ + 
—i— — &c.)” this series is capable of being expanded into 
another, which also proceeds according to the even powers 
of 0 ; first let k = 0, then comparing this with (A), we have 
\ = (cos 0)” — (cos 20)” + (cos 30)”— &c. (3,9) 
Let us now examine if the definite integral is finite, it is in this 
case. 
fdv{ 
cos 
p 0 1 J_ 
0 I | 1 T n r 7 I rji ^ v 
— -}=I dv L +_ 
J 
=/*(- f-’ Y 
Z n 1 0 “ I 
1 i — n - 
i + {2—n) 
+ 
n ,n—i 
1.2 
i + ( 4 -«)- 
7 T 
+ &c -l [:=:] 
If n is a whole positive number we already know that the 
series (3,9) is correct; if n is a fraction the series which ex- 
presses the value of the definite integral is finite for all values 
of 0, except such as are contained in 0 = — i being any 
whole number; if n is negative, then the series is finite for all 
positive values of 0 ; it appears then that whatever be the 
value of n if0 is positive, we have 
