3” 
+ 
Let a— 3, b—m—r— i; then-- 4- 
I • 2 *3 
„ 1 ,35 
4 &c.=-X 1 + 
1 2 24 
I » 2 
Let a=o, i=i, m=i> tlien 7 ~ + 3T5T 
_ 3 1 grc. —- Vid. Phii'ofophical Tranfa&ions, vol. 
5-7-9 8 
lxxii. page 389. 
Ex. 8 . Leti4-^ 4 ^- 4 -T 77 + &c. be a defending 
' b bd baj 
feries, and fuppofe 
ALGEBRA. 
Let a*— 1=0, then x=:i> or—1; if x—1, then 
5 6 
J | r _ . 
2-3-4 3-4-5 
1 2 . 2 1 2 o 
.— i-1-j--j-&c. — o 
2 1.3 2.4 3.5 
Tiiat is, ——j- 
1.3 2.4 
■+i + M+«7' +&c -= s 
Then 
are aces 
b'bd' bdf ' bdfh 
■ See. —S —1 
n r . ci a c ac c _ _ 
By fab. ,- ? + ? x.- S + S X.-^ + «=c=" 
_ b—a a d—c , ac f—e 
+ (XT+« >! T + =' 
3-5 2 
And _i- 4 ._L. 4 - —-U &c. = - 3 -. 
1.3 2.4 3.5 4 
If x— —i, tlien 
I 2 2 . 2 „ 
—i 4-f----4-.— &C.rs:0, 
' 2 i -3 2 -4 3-5 
And —-1- See. - 1 . 
1.3 2.4 3.5 4 
^ 3 ^ 
Ex. 3. Let i4-1-4-U &c. =S, andmultiply 
2 3 4 
both tides by 2 x—1; then 
—1 4_hi 4 L—U L_^_ &c. =2x—1.5; let ix —1=20* 
1.2 2.3 3.4 
or a'—-, and 
2 
Aflame a = n 
c—n-\-i 
e—n- j-2 
Sec. 
m—n n m—n 
Then- 1 -X —;— 
m m 7/2 —|— 1 
b—m 
d—m-^i 
f—m+2 
Sec. 
n.n-ir-i.m—n 
' - 4 Sec. — 1 : or 
Or 
1.2.2 
3 
r + 
2 . 3 .2 * 
4 
3.4.2- 
5 
■ &c.=o, 
4 & c - =1. 
n 
n.n-\-i 
m.vi 41 ,m-\-2 
2 , 2 -3 
4&c.— -; ifn—2, then 
_74- Sec.—~ 
m m.m-\~\.m-\-2 . m 
m—n 
1 
— 2 
Similar to the method of fubtraftion is the following, 
given by De Moivre, Mifcell. Anal, p.130. Adame a 
feries whole terms converge to 0, involving the powers of 
an indeterminate quantity x; call the fum of th.e feries 5 , 
and multiply both tides of the equation by a binomial, tri¬ 
nomial, &c. which involves the powers of x and invariable 
coefficients; then if x be fo adiimed that the binomial, Or 1 
trinomial, &c. may vanith, and fome of the fird terms be 
tranfpofed, the fum of the remaining feries is equal to the 
terms fo tranfpofed. 
v y2 y 3 
Ex. 1. Let 1-j--f- --f- Sec. z=.S 
234 
Multiplying both tides by x—1, we have 
1.2.2 2.3.2- 3-4-2 3 
If both tides of the equation be multiplied by a bino - 
mial, each term of the feries obtained will have two fac¬ 
tors in its denominator; if by a trinomial, each term will 
have three factors, &c. 
X .^3 
Ex. 4. Let i-|-j-7-f- Sec.—S, 
2 3 4 _ 
Multiply both fides by 2x — i.x —1, or 2X 2 —3x41 $ 
then 
2X 3 2X 4 
2A 2 4--j-1- &C. 
2 3 
1 —3*— -- -- &c. 
2 3 4 I J ‘ 
x x 2 x 3 x 4 I 
+*+ 2 Jr T + T + y + &c - 
3 A J 
3 * s 
5 * 
5 * 
,-2 
6 x 3 
7 X 
,-4 
I.2.3 2.3.4 3.4.5 
If x=j y then i— -4--— 
2 1.2.3 
< 6 
And 
J 
— 2\ 2 -3A4l.S 
2.3.4 
4 &C.r=0, 
1.2.3 2.3.4 
5 . 5 1 
- - —1 .S 
Or- 
X* , X° , X* 
*+T + 7 + T + &c - 
X X 2 X 3 X 4 | 
2 3 4 5 J 
X X 2 X 3 - 
-14 -i- 1 -h &c. =x— 1 . 5 ; and if x~i, 
1.2 2.3 3.4 
— 1 * 
4 1.2. 3 -2" 
—--h&c.tzr-. If x=-, then 
3.4.5 2 2 
c „ 
4 S &C. —o, or 
2 - 3 - 4 - 2 ° 
3-4-5- 2 
1.2.3.2' 
Ex. 5. Let 
2 . 3 - 4 - 2 J 
I 
3 - 4 * 5-2 
v2 
+ ” 
■ Sec. ~ . 
4 
4&c.=5; 
then —14——I--L4L4 Sec. 
1.2 2.3 3.4 
-o; or-1- \- 
1.2 2.3 
3-4 
4 Sec. 
m m-\-r ‘ ?»4 2r 
Multiply both tides by the binomial ax — b ; then 
— 4 4 &c.l _ 
m w-\-r 7»4 2r \.=zax—b.S, 
h bx bx 2 bx 3 _ ( 
&c. 
Ex. 2. Adlime j-j-J-j-p. & c . and multi. 
2 3 .4 
ply both tides by x 2 — 1 ; then 
y 3 y>4 
* , 4 t 4 - 4^-1 :_ 
x x 2 x 3 x 4 /-=:a 2 1 X 5 , 
--&c. 
2 3 4 5 J 
_ X 2 X 2 , 2 X 3 . 2 X 4 , , - 
Or —i-!-1-1-[- &c, “x B —i X 5. 
2 1.3 2,-4 3.5 
Or 
m-\-r m-\-ir m-\-3r 
I; m^-r.a—mb 
m 
—ax — b.S. 
Let ax — b—o, and tranfpofe 
m^-r.a—mb 
m-\-ir.a —ra4 r -^ 
X x 4 — -- — X 4 Sec, 
m+r.m+zr 
b , 
-; then 
. ■mA-2r.a—viX-r.b h 
X * 4 -4-L— Xx2 + &c . 
?« 4 r.m 4 2r m 
