412 
Mr. vince’s new Method of 
For let 
x 4 nz 
w-\-nv 
X -j— /< -p u a 
w -f - /: -f- a 
--be an? two fucceflive termsof the 
. v 
S^ 
veil feries, which, if we begin to colled! at the fir ft term, will he 
the two general terms to be collected into one, and will therefore 
give : — — tor a general term of the relultmg feries. 
w 4 nv x zu 4 ft 4' a • v 
Let ns now make » infinite and thenthis term will vaniftgandcon- 
fequently the refulting feries will terminate at an infinite diftance. 
Now, by making n infinite in the given feries, the two fuccef- 
iive terms 
(as they begin with 
x 4 rrz 
ZV + HV 
x + n.+ a . % 
w 4 n 4 a . v 
by making n 
infinite) become % - - ; this feries, therefore, is ftill continued 
' V V 
after the other terminates ; and the terms of fuch a continua- 
tion will be 
V V V V 
■4 &c. ad infinitum, and whole fum 
by cor. 2 . to the lem. is ; confequently the given feries ex- ' 
ceeds that which is formed by collecting two terms into one, 
beginning at the firft, by -L ; hence the fum of the latter feries 
• /* 
4- — will be equal to the fum of the former. Now, if we begin to 
colled at the fecond term, then will -Lh— q. ~- + n _fd ' 2 be two 
w-\-nv w-\-n-\-a . v 
general terms of the given feries to be collected into one ; con- 
lequently the continuation of the given feries, when n becomes 
infinite, will be + - -&c. ad infinitum, whofefum 
V V V V 
by cor. 2 . to the lem. is - — ; in this cafe, therefore, the fum 
of the given feries is lefs than the fum of the feries formed by 
collecting two terms into one, beginning at the fecond term, by 
