finding the Value of an infinite Series , f ee. igx 
that fix or eight of its terms will give the value of 
the whole (and confequently that of the original feries 
a-bx+cx x-dx^ + ex^- fx^+gx^-bx 1 + &c. to which it 
is equal) exadt to feveral places of figures, even in the 
molt difficult cafes : for if x is = i (which is its greateft 
pofiible magnitude) i+x will be — i + i or 2, and confe- 
quently t Tx[i i +a , ) 3 5 1 +A'tfi 1 +x) ; , and the following 
powers of 1 + x, will be equal to 4, 8, 16, 32, and the 
following powers of 2 ; and the powers of the fraction 
X 
ppp will be equal to the powers of Therefore the feries 
a 
bx 
D T XX 
D 11 * 3 
TTI A .^ D IV ^. 5 D v * ( 
D V *X‘ 
1 +■*■ r+i* r i~t xV 
in this cafe be = to a 
I +a 1+ I + h’ I -f d 
b D 1 D 11 D IXI 
I-M 
8cc.will’ 
D IT D T E VI 
~fi~ 64 - 778 -tZC ' 
4 s x6 
the terms of which decreafe in a greater proportion than 
that of 1 to 2, becaufe. the numerators a , b, d 1 , c", d 111 , 
D 1V , d v , D VI , &c. form a decreafing progrefilon, and the 
denominators increafe in the proportion of 2 to 1 . 
Of the invejtigation of the foregoing differential feries. 
Art. 5. The foregoing differential feries was invefti- 
gated by firft, fuppofing the original feries a-bx+cxx 
-dx i +ex*~fx s +gx (, —bx 1 + & c. to be equal to another 
feries whofe terms fhould involve the fame powers of .v 
as the former, but in which every power of x fhould be 
multiplied. 
