the Value of a slowly converging Series. 189 
which three parts, for the sake of reference, may be called A, 
Bx 8 , and Cx 16 . It is evident also, that the numerical values of 
the first eight powers of x are all that are required for finding 
the sum of these twenty-four terms of the series proposed : for 
Bx 8 + Cx 16 is = x 8 (B -j- Cx 8 ); and this product A = the 
sum required*. 
The numerical values of the first eight powers of x are 
these : 
x = o*9, 
x*= o*8 1 , 
x 3 = 0729, 
x 4 = 0-6561, 
X 5 = 
x 6 = 0-531441, 
x 7 = 0-4782969, 
x 8 = 0-43046721,; 
from which values of the powers of x, the values of A, B, and 
C, continued to twelve places of decimals, are easily obtained, 
and are as follow. 
* The same thing might have been obtained from the first four powers of x, or from 
the first six powers of x ; as might likewise the sum of 32, 36, 40, or any other num- 
ber of terms that is a multiple of 4 or 6. But the number 8 is here chosen, for the 
sake of a subsequent use that will be made of the value of x*. 
