Mr • Vince on the Sums 
'i ~ + h ~ i +&c * ad in f ‘ 5 we have > b y the 
300 
As —4= 
x 
fame method of proceeding, A-B-fC — D 4- &c. ad hf = \ ; 
confequently A + G -j-B + &c. = f, and B 4- D + F + &c. = j. 
Becaufe 
X I X iV 
+ -5 +— + &c# ad inf. ', if for x we 
write 2, 4, 6 , &c., then will — 
1.2 3.4 5 
1 1 
f- + &c. = (Tab. 
&c. = 
3) A // + B // + C // + D" + &c.; but _L + 
1.2 3.4 5.6 
hyp. log. 2 ; hence A" + B" + C" + D" + &c. = hyp. log. 2. 
If in the fame expreffion we write 3, 5, 7, &c. for x, then 
+ 6~i + &C ’ = ( Tab - 4 ‘) ^ / + ^ // + c ' / + &c. ; but 
2.3 4.5 6.7 
I 2 I 
// 
~ + ^ + + hyp. log. 2; hence a"+b"+c 
+ &c. = 1 - hyp. log. 2. — Hence from either of thefe two laft 
cafes, we have a very expeditious method of finding the hvp. 
log, 2. 
prop, ir, 
‘To find the fitim of the infinite femes whofe general term is 
mx dtzn 
By divifion — 
n~ n 
— i— 
nix' rb n mx r m % x Xr m 3 >x y rnTx 
4 v 4 r 
4- & c. ad inf. 
hence, if 
mx r dzn 
be made the general term of a feries, and for 
x we write 2, 3, 4, &c., its fum will be equal to the fums of 
another fet of feriefes, whofe terms are the powers of the 
reciprocals of the natural numbers refpeclively multiplied 
into 
