SERIES. 
2 _ 
3 ^’5^ 
4 6 8 „ 
X. - X, &C. ; 
6 ^'’ 20 
42 
1.1 3.1 
2 . 3 ^’Trs’^’ 
'- JLm 
7 ’ 9 
and therefore the terms may be continued 
indefinitely by the successive multiplication 
by these fractions. Also in the following 
series 
1 .r -j — — -| — ^ x’ -| — xK &c. 
'6 *40 ^120 ^1162 ’ 
by dividing the adjacent terms successively 
by each other, the series of quotients is 
1 9 g.l 49 
■ X, &c. or 
5 . 3 7^ 
b. 7^’ 8.9 
and therefore the terms of the series may 
be continued by the multiplication of theSe 
fractions. 
Series, surmnation of. We have before 
seen the method of determining the sums 
of quantities in arithmetical and geometri- 
cal progression, but when the terms in- 
crease, or decrease, according to other laws, 
difierent artifices must be used, to obtain 
general expressions for their sum. 
The methods chiefly adopted, and which 
may be considered as belonging to algebra, 
gre, 1 , The method of subtraction. 2 , The 
summation of recurring series, by the scale 
of relation. 3. The differential method. 
4. The method of increments. We shall 
content ourselves with an example or two, 
in the first of these methods. 
“ The investigation of series, whose sums 
are known by .subtraction.” 
Ex.i. Letl+1 + i-j-l+, &c. in 
inf. = S, 
Ex. 3. Let ~ — L^1L, j — 1 - 4 . in 
1.2^2. 3^ 3.4^’ 
inf. = S, 
’ \ 
by subtraction, — 1 ® 
’ t.2.3T^2.3.4^3. 
- 1 -, &c. in inf. z=; 
and 
1 . 2 . 
inf. = 1 . 
4 
2.3.4 ‘3.4.5 
4 . 5 
S &c. in 
Ex. 4. Let ^ 
2 
m -(- )i — 1 . r ' -f- M r 
then — 1 1 1 4 _ 
m-^-ar ^ j»-4-3r ' 
_i__S 1 
nt-j-nr ‘ tu’ 
by subtraction. 
1 
+ &c. (to n terms) -j- L — _ . 
wt-j-wr m’ 
hence, — 
m . m + r ' wi + r . m + 2r 
(to n term.s) = ^ 
“I- &c. 
and 
tfi ' m -j- r’ 
then 1 -f 1 -f 1 4- 1 -f, &c. in inf. = S- 
- 1 , 
m . m -j- »' /« -j- r . »j -j- 2 r 
(to » terms) — -1 ^ 
m r m r -j- n ' 
If n be increased witijout limit, - 
h ) 
by subtraction, ~ — [- — 4 - — 4. 
^ ’ l.2“2, 3^3.4 • 
'in inf. =a 1 : or 1 -4- l-U- — 4- — ,&c — 1 
’ 2~6‘ 12^20’ 
Ex. 2 . Let 1.4-^ + 2 + 5+j 
inf. 3= S. 
^en I -f 1 -f- 1 1 +, &c. in inf. = S— f, 
by subtraction, ^ + + 
&c. in inf. = - ; 
2 ’ 
111 . 
or U -4- 
1.3 2.4 ‘ 3.5 
inf. = |. ' 
J« r -J- n I-' 
vanishes, and the sum of the series is 
m r 
If m = )• = 1 , we have — 1 1 - —1- 4 . 
1.2 ~ 2.3 ^ 
3.4 
4 &c. (to re terms) = 1 
1 -f- re 
1 I 1 1 . 
+ 4^6 +' 
1 -j-re" ' 
Similar to the method of subtraction is 
the following, given by De Moivre. 
“ Assume a series, whose terms converge 
to 0 , involving the powers of an indetermi- 
nate quantity, a: ; call the sum of the series 
S, and multiply both sides of the equation 
by a binomial, trinomial, &c. which in- 
volves the powers of x, and invariable co- 
efficients ; then, if x be so assumed that the 
binomial, trinomial, &c. may vanish, and 
some of the first terms be transposed, thp 
