3'° 
Mr. Vince on the Sums 
Cor. Hence, as i + — + -7 +— + &c. = 2, we have i + I + 
3 6 10 6 
~ + &c. = 2 hyp. log. 2, and — + — + - + &c. = 2 - 2 hyp. 
15 3 10 21 yx 
2. 
’F R O P. X. 
Ifb find the fum of the infinite ferles 
i , 1 1 „ 
1 + + &c, 
4 10 20 
icing the reciprocals of the figurative numbers of the j\.th order 9 
having the figns alternately + and — „ 
If we write 2, -3, 4, - 5, &c. for the general term 
will be ~ — = 4-4-4- + 7 \ + &c. ; hence the fum required =* 
6 b + 6 d+ 6/+&c. = (by Prop. 3.) - ?\ + 12 hyp. log. 2. 
Cor. Becaufe the fum of 1 +~+ — + — + &c. = C. • there- 
4 IO 20 2 
fore 1 +— + — 4- &c. = — 3 + 6 hyp. log. 2 ; and — + — + — 4 . 
10 35 0 Jk * 4 io t s 6 
&c. = 41-6 hyp. log. 2. 
p R O P. XI. 
c Io find the fum of + & c . ad infinitum. 
1 • 3 - * 4 3-5 
The general term, by writing 2, 3, 4, &c. for x, is 
X 2 * I 
+ 4 +& c »l hence the fum = A+2C + 2B 
I XAT+I * x* ' 
-J-&C. = 5884966993407, 
P R O F. 
