492 ON THE SUMMATION’ OF SERIES, 
This remarkable series, which expresses the 
quadrature of the circle, may be made to have 
any degree of convergency, by assuming differ- 
ent values for y. 
If we take y=6 .. 4y—3=21 and 4 y—-1=23, 
and consequently equation (22) will become 
7 6 1 1 23 J 3 
Tang ae (ax deny) + aie (+) wn 
1 21 
1 l 5 1 1 
(sr- a5) +74 ( op = )- &c., &e. 
=.7646006 +.0222309 — .0015528 +.0001119= 
-7853989 
Hence it appears that by taking only two 
terms of the complement, together with the terms 
under =, we have the quadrature of the circle 
true to the sixth place of decimals. 
Ex. 6.—Let it be required to sum the series 
1 -~54+4 -~ft+e- + &e., to infinity, which 
is the hyperbolic logarithm of 2. 
The general term of this series will be, evi- 
dently, <4 — al consequently, we shall have 
