of Infinite Seths-. 305 
. } &c, - - b - JL-j -2 hyp. log. 2. In like manner the 
■ 3 • 4 3 • 5 4 
fum of 
+ 
2 . 3 s • 4 3 • 4 5 • 5 
&c. = — £ — d - $■ 4 - 
4 
2 hyp. log. 2. Thus we may -proceed as far as we pleafe by 
adding two powers to the middle term; and hence this remark- 
able property of the feriefes, that the difference of the fums of 
the feriefes where the middle term is x, x s , x . &c. is b , d \ f y 
&c. refpecfively. 
j 
Ex. 2. In like manner if the general term be ==— , 
and we write 2, 3, 4* See. for x, we have - - — . ^ 
'j l 4-&C. SD4-F4-H+&C. = (by Prop. i.)~ - B. 
' 3 • 4 s - 5 4 
Hence alfo - +&c. = ^--B-D; and fo on as 
l , 2, . j ^ • Of “T 
before. 
If the general term be under the form - ■== * 
will be moft convenient to refolve it thus : by divifion 
JL® I -2L+5L - &Ci 
xA-m x x z x 
m 
xfi-m 
hence 
x + m 
i JL+1 
x + m x x z 
m 
4- &C. X — ■= 
m 
x . x + m 
, m my « I 
xx m 
where the fign on the left hand will be -f- or — according as n 
is even or odd, and the number of terms on the right is = n. 
Now the fum of the feries whofe general term is j— -== i® 
well known, and the fums of the other are found from the 
tables. 
Ex. 1. To find the fum of -+~r— A +^~ + & c * a ^ m f‘ 
Yol. LXXXI. 
2-3 3-4 4 
S f 
Here 
