invefii gating the Sums of infinite Series, 4 1 1 
ienes ; but — • \- — ~ + —— 4- &c. is equal to the hyp. log. of 2 ; 
therefore the fum of the given feries is = — - 4- hyp. log. of 2, 
the fame as before. 
case 2. Let the given feries be - 
- + 3 + - 4- &c. 
5 7 9 
Here r=2, 77=1, 2, 3, 4, &c. m — 1. Now, if we 
begin to collect at the fecond term, the feries becomes 
— h_4- -— 4 1 — p&c. and the corre&ion, to be fubtrafted . , 
5-7 9 • 1 1 
being; - , we have — 4- — — 4 - — f &c. - - for the fum of the 
0 4 1 • 3 5*7 9 • 1 1 4 
given feries ; but— — p_ 4- — - — 4- &c. is equal to a circular 
1 - 3 5*7 9 • 1 1 
arc (A) of 22°i, whofe radius is unity; therefore the fum of 
the given feries =A- 
PROP. II. 
If x + r ‘ % - he the general term of a feries formed by writing for 
w + nv 
n any feries of numbers in arithmetic progrcfjion , and whofe terms 
are alternately 4- and - ; then if a feries be formed by collecting 
two terms into one , beginning at the firfl term , the fum of the 
feries thence arijhg will be lefs than the fum of the given feries by 
If a feries be formed by beginning at the fecond term , the 
2V 
fum thereof will be greater than the fum of the given feries by 
H h h 2 
For 
