w-f -a 
4 1 o Mr. v i n ce*s new Method of 
colled at the fecond term, then will - + 
. be the 
rn + m n + a . r + m 
two fucceffive general terms of the given feries to be collededinto- 
one; confequently the continuation of the given feries when n be- 
comes infinite will be - - 4 1 - I 4 - - &c. ad infinitum, 
V V 1* v 
whofe fum, by cor. i. to the lem. is- — ; in this cafe, there- 
2 r 
fore, the fum of the given feries is lefs than the fum of the 
feries formed by colleding two terms into one, beginning at 
the fecond term, by -L ; hence the fum of the latter feries 
- ^ will be equal to the fum of the former . 
Case i. Let the given feries be - — ~ 4 2 _ - -f &c. 
2 3 ^5 
Here r= i, n= i, 2, 3, 4,&c. and m- 1. Now, if we begin 
to collect at the fiifl term, the feries refolves ltfelf into 
“ ~ 6 ~j “ ^ C ' anc * t ^ ie corre ^ on > to be added, being 
2 9 we ^ ave “ r~-7^7~ t~ : -&c. +- for the fum of the 
• 3 4 • 5 0 • 7 2 
given feries. Now ~ &c. is well known to- 
be equal to — 1 + hyp. log. of 2 ; confequently the fum of the 
given feries is = - -h 4 hyp. log. of 2. 
If we begin to colled at the lecond term, the feries becomes 
77 z + + J76 + &c * and tlle corre ^ioi^ to be fubtradled, being 
— , we have — - + + 4 &c. - x - for the fum of the given 
feries ; 
