4 r 5 
invefii gating the Sums of infinite Series, 
Befidcs the feries contained in the foregoing proportions, a 
great variety of other feries might be produced where a cor- 
rection is neceflary, after collecting two terms into one, in 
order to exhibit the true value of the given feries. As the 
proper correction, however, may always be found from tire 
principles delivered in the above proportions, that is, by confi- 
dering what the terms of the given feries become at an infinite 
diftance, 1 fhall only add one or two inftances more, and con- 
clude what I at prefent intend to offer on this fubjeCt. 
ex. i. Let it be required to find the fum of the infinite fieries 
3 i 4_4 j J + 5 1 6_6._ I &c> 
1 - 2 2.3 3.4 4.5 
l6 
This, by refolving two terms into one, becomes - — - — - +■ 
&c. ; and as the terms of the given feries 
24 
+ 
32 
3 • 4 • 5 5 • 0 • 7 
continually approach to unity, the correction, to be added , is 
- , confequently -- 1 ' — + -■ - 2 ^ - ; + — 3— . _ &c. + -f is equal to 
the fum of the given feries ; but by prop. 1 . part I. the fum of 
the feries — — 1 + t~i~: + & c - is ec l ual t0 83 ~ 2 (S 
1-2.33.4.55.0.7 
being the hyp. log. 2 .) conlequently the lum of the given feries 
is 8S — 1 §. 
ex. 2 . Let it be required to fi/rd the fium of the infinite fieries 
ill _ L'Jl 4- ill - ili + &C. 
1-3 3-5 5-7 7-9 
This feries, by refolving tw r o terms into one, becomes 
+ &c. and as the terms of the given 
feries 
+ 
8 
+ 
1 - 3-5 5-7 -9 9 • 1 1 • J 3 
