336 NEW SERIES for the 
m (wt ge ee) es 
6 (x — 1)* 8.9.10 
I =J+4 I es SF pate i X'+12X’—13 
_ log* x | 316 X+- 4X4 3 310° X'+4 X74 3 
| ne ee 2 XO TS 
Loh grote tes ee 
The terms of this feries approach continually to thofe of a 
: 5 f Kren/ tad 
geometrical feries, of which the common ratio is OE and 
hence it follows, that the fum of all the terms after any aflign- 
ed term, approaches the nearer to ra of that term, according 
as it is more advanced in the feries. 
65. BesipEs the foregoing, our method furnifhes yet another 
kind of expreflion for the logarithm of a number, namely, a 
product confifting of an infinite number of factors, which ap- 
proach continually to unity. Such an expreflion may be inyef- 
tigated as follows. From the identical equation 
X r= (XK? — 1) (KF $0). 
Let there be formed the feries of equations 
ho I =2(e? 2th 
2@?—1)=4e* ete 
z 
-_ m 
“ (x"—1)=m (x” —tI) pM Bis = a 
here 
