Here the generaltermis — — L== i= 4- , and by writing 
X X -f- I X , X L X 
2, 5. 4, &c. for x, we have the fum = - — * &c. 4 - 
J 2-33-4 
A = — - -f A. In like manner — + - 4 — + — i- + &c. = - i + 
a 2 3 . 3 4 - . 5 6 . 7 
hyp.log.z + A". Alfo^i- + + + &c.=i|-3A + 9 Bx. 
I? 
2? # _ 
If m be negative, then — - _ = — — — — &c. x i*. 
Hence -r^— + + — — + &c. = i — A - B - C ; and fo on 
2 4 . 1 3 4 . 2 4 4 . 3 
for others of the fame kind. 
If the general term be under this form L=^, then, in 
° rn n , 
x . ox -\-.m 
like manner, we have =*= 1 - = — — + — & ci. 
x" . ax’+m ax"+m ax' *V" 
CL r 
[x_, where the fign on the left hand will be 4* or — , accordr 
in 
ing as r is even or odd, and the number of terms on the right 
is = r + i. 
Ex. t. To find the fum of ~ — |-^ b — b &c. 
2 - 5 3 4 • 10 4 • J7 
Here m = r, n - 2, r — 2, a = 1, and the general term — — — 
* 4 X* 2 +i 
!*=, — L_— 1 + — ; now the fum of the feries whofe general 
1 X 2 -1- i X 1 X 4 ° 
term is ^r~ is = ,576674037469, by Prop. 2. ; confequentlj 
the fum required = ,576674037469 - A + C = ,014063204532. 
Ex. 2. If the given feries be —i_ -p~L- + ■■ ■ - 1 - ■ + &c. the 
general. 
