Infinite Senes-* p t 
+ &c. &c. reduce the fractions to a mixed number, fo that the 
dimenfions of % in the numerator of the fraction be lefs than 
its dimenfions in the denominator, and the integral part be 
Ae % -f- Bze* + CaV® 4- Dz*e* + . • . llz ltH e z : the fum of the infinite 
feries whofe term is Ae* + B ze* -f Cz z e % -f D z*e* 4- E% V 4 . . „ 
4 - Uz m e % 
(' • 2 • 3 
V (i--?) 4 
he 
- I , 
I — s \ ( 
I+2XI .2 
De 
(ffg) * Eg ( ' ~ p-«F 4 ~ &c ) : thefum 
of a feries Y whofe general ter m is z m e % )= - - — - 3 ‘ 4 " g — 
(!_,)”+« 
JgWf—I IgW 1 ^ TgOT 2 2, W 1 
—pi . 2. 3 ..»-if + ' X 1. 2.3 . m—z . g 
; • ' 1.2.3.. m — h—i ... where Lis 
equal to the fum of all quantities of the following fort, 
“S’” -1 x x y S’"~ r, ~P~ l x i S’" — 1 ®— /— 1 x ‘S’” — “ — z 3 — > — ■* — 1 x &c. 
where «, /3, 5/, $, &c. are whole affirmative numbers ; (in the 
preceding notation by 'S? is defigned the fum of the contents 
of every w of the following numbers 1, 2, 3, 4, 5, . . . ^ . 
and + 1+« + &c.=^+i ; the above-mentioned pro- 
duct “S’" -1 X ^S” - ““ 1 X &c. is to be taken affirmative or nega- 
tive, according as the number of letters a, fi, y, $, &c. is even 
or uneven. 
The fum of the feries z’” X e~ may alfo be found by affirming 
for it (az m + bz’"-' + cz’*-* + dz m - s . . . k)e % ; then, finding its 
fucceffive term (a x 2 + 1 <? + <fo+i *e + cz+ 1 *e + &c.) e* } 
and taking the difference between it and the affumed quan- 
tity, there refults (axe — n*+ nme + be - 1 z m — 1 + &c.) e x ; 
by equating it to the given term z m e* are deduced the fub- 
fequent equations axe-i — i, mae+be- 1= 0, &c. whence 
N 2 a=z 
