invejligating the Sums of infinite Series. 39 1 
Now the (urn of the lower feries, omitting the hrfl term, is 
2 S — 1 
equal to — B divided by r % or = — 2 - ; hence, by tranfpo- 
lition, and, multiplying both Tides by h, we (hall have, 
b lb 0 b S , 2/;S — r+ I . b , 
+ &c. .. = — \r 2 ; alio by 
r + I . 2 r-\- I 2 r -x- I . y-\- 1 
I 
multiplying B by - we have 
a 
+ 
- &C. . . • = 
2 a S — o 
; fubtra£t 
I . r + I r + 1 . 2r+ I 2 r + 1 . 3 / + I 
the laft equation but one from the laft, and we ihall have 
a + b + a + lb 
- &c. . . . = 
'iro— r + 2 .ixS-rfl + r+l . b 
1 . r + I r-t- 1 . 2./ + 1 2 r + 1 • 3 r + * 1 
Cor. i. Hence it appears, that the Turn of this feries can 
never be exhibited in finite terms, except a : h as r 4- 2 : ar, in 
which cafe thefum is equal to 
r -f 2 
5 
Hence, if# = 2,thenr — 1 ; 0 . 4 
See. . . . — 1 ; 
i* = 1 ,b = 4, then/- = p, A jf ? ~ It ■ „ . IS 1S . ,, 
6. . 4 7 !i_ +kc ....=2, 
1 • • _ ~ . _ _ _ 1 . „ „ „ 
5 
9 + T 3 
J 7 
-J- &c. 
I 
6 
= 4,3 = thenr = , 7 . 23 . 23 . 19 
20 
Cor 2. Put a~c -b, and we fhnll have, after tranfpofition, 
c c 4- b 
--f 1 . 2>4r 1 ar+i • 3 r +‘ 
~P Sec . ... — 
3 ' + 
o . b - 2 rc X S - 2 r 4- I b+rc _j_ c-» 
r-t- X 
prop. I. 
To find the fum of the infinite feries - - : f. — ; 
+ 
m + n 
m 4 - 2 « 
v 
-f 1 . 3/' 4- 1 . 4 '-+ 1 4 r + 1 • 5 r + 1 * &■+ 1 
+ &C. 
Every 
