2 I , 3 - 
-X- 4 --X- 2 
J -3 3 3-5 3 2 
, I8 A L G E 
If the terms of this feries can be made to coincide with 
the terms of a propofed feries, the fum of the latter may 
be found. Thus, let the fum of the infinite feries 
4 - Sec. be required. 
5-7 3 3 
In this cafe a=-, therefore-— b—o, or a—ib\ alfo 
_3 3 
vi—i, r—2, and m-\-r.a — ml>— 2, that is, 322— b, or 9 b—b 
— 2; hence b—~, a—-- and, if thefe values be fubftitu- 
4 4 .... 
ted in the fucceeding terms, the general feries coincides 
b 1 
with the propofed one, whofe fum is therefore —, or-. 
After the firft n terms the feries becomes 
v 4-2 1 , ?2+3 
: X ,, , 1 t-— 
2 n-\-i . 2224-3 
1 , 0 1 
, _-x-b&C.or— X 
3 2224-3 • 2,; +5 ■* 
22-f-2 1 224.3 1 . „ , , 
- - X-4- ■ - X—4- Sec. and, by com- 
zn-\-i . 2n -j-3 d 2«-j~3. 3 “ 
1 
paring this with the general feries, we find x—j, an( l 
27—32$; alfo jn—m-%-1, 7 — 2, 77z4-r.fi— mb—n-\-z, or 
2 ”*!"3-3 b —2224*1 -b=z 224-2 ; therefore a r n-) r 1,.b—n-\-z, and 
, 1 , 3 224-2 , n-\- 3 
b—-, a—2b —hence -—- X 1 4- - 1 - 
4 2724-1.2«4-3 3 2724-3 - 2,l +5 
i _ b 1 , 1 
x — 4 - &c. = — = and — X 
3 “ m 4 . 222 -J- I ^ 
72 - 1-2 
:X- 4 -- 
«-f 3 
:X^- 4 -&C- 
2224-1.2774-3 3 272 + 3.222 5 3 4.2724-1.3 n 
the fum of the feries after the firfi: 72 terms; alfo the fum 
* ' .1 
of the whole feries. istherefore the fum of the firft n 
4 
1 1 
terms is- — —. 
4 4.2224-1.3” 
Cor. Since the fum of the feries after the firft n terms 
1 
4.2224-1.3" 
1 1 
'4.1.3 0 — 4' 
if n— o, we get the fum of the whole feries 
28 1 
~Xi + 
Ex. 6. To find the fum of —— X- 4 - 
1.2.3 4 
39 1 52 1 . „ . 
-X—t 4 --X- 1 - Sec. in infinitum. 
3.4.5 16 4.5.6 32 
Becaufe the fa&orsin the denominators increafe by 1, 
^ X x^ 
and begin from 1, aftiime i 4-+ — + — 4 Sec. 
2 3 4 5 
axS, and multiply both fides by ax" — bx-\-c\ then 
B R A. 
1222—8&4-6c=28; from which three equations it appears, 
that a— 6 , bzz-j, e—2; and, if thefe values be fubftituted 
in the general feries, we have 2—34-—— X - 4 -———X 
1.2.3 4 2.3.4 
1 39 1 52 
-4 -——X— 4--~—7 
3.4.5 16 4.5.6 
28 1 1 39 
-X-+ : 
2 - 3-4 8 3.4.5 
1 1 o 19 1 . 
X — 4 -&c. =0, or— Z-x- 4 - 
32 . 1.2.3 4 
16 
4- &c. — r. 
On Recurring Scries. 
If each fuccdeding term of a decreafing feries bear an 
invariable relation to a certain number of the preceding 
terms, the fum of the feries may be found. 
Let a-\-bx^-cx*^- & c - be the propofed feries; call its 
terms A, B, C, D, Sec. and let C—fxB-ygx i A, D—fxC-\-gx*B, 
&c. where f-\-g isthe fcale of' relation ; then, bythefup- 
polition, 
A — A 
B — B 
C — fxB 4- gx*A 
d — yivC 4- 
F. — fxD 4 - gx'C 
See. 8 e c. 
and if the whole fum A-\-B-\-C-\-D-\- Sec. —S, we have 
S—A-\-B-\-fx X S— A+gx^xS, or 
5 •— fxS — gx'S — A 4 - B — fxA ; therefore S — 
A+B-fxA 
i—fx—gx'' 
In the fame manner, if the fcale of relation be f-\-g^- 
See. to 72 factors, the fum of the feries is 
A-\-B-\~C — (n)-fxxA-\-B —(tz— 1)—ffx s X A-\-...{n-2) Sec. 
x—gx*-kx 3 .(224-1) 
Ex. r. To find the fum of the infinite feries 14-2*4- 
2x , -^-4.x 1 -\- Sec. when * is lefs than 1. 
Here f~2. g— —1, and the fum 2 =^———z ~ 
1—2A-J-X 
If x be equal to or greater than t, the feries is infinite; 
yet we know that it arifes from the diviiion of 1 by 
i—x f, and the fum of n terms may be accurately deter¬ 
mined. 
The feries after the n firft terms becomes »-j-i.*” 4 - 
224 - 2 .x’ ! +' 4 - 224 " 3 -' ,,! + !! 4 - See. in which the fcale of relation, 
as before, is 2—1 ; and therefore it arifes from the fradtion 
224-1 . X n -\- n -\-'2 • X n ft '-2.22-(-I .X ?i + ‘ 224-1 ■ X V - 72 V'”+ 1 
I — 2 X 4 -a‘ i i—X ) 3 
therefore, r 4 - 2X 4- 3*' 4- Sec. to n terms, 
1—224-1 .x”4 - nx n A-i 
- ax 3 
ax 2 4 - 
2. 
2ZX 4 
+“T 
-f- &c. 
bx 2 
bx 3 
6 . r 4 
Rrr 
2 
cx 2 
3 
ex 3 
i.__ 
4 
C.V 4 
j _ 
CtL. 
\ . Zrr 
4 
+ 5 
OC b • 
cx 
+‘ + -~+ 
2 
_ , c—ib , 6 a —364-2C i2fl— Sb- 4 - 6 c 
Orc-{--—— x *4 - TVT 1 X A 4 - -—A L-Xx 3 J r 
1 —xr 
Ex. 2. To find the fum of 22 terms of the feries 14-3* 
-|~ 5 * e 4 - 7 ^ 4 " Sec. Suppofe f~\-g the fcale of relation; 
___ then 3/+.f—5, and 5/4 2g=l ; hence/=2, and^=:—1 ; 
Sec. ^—ax* _ bx -f c X-S> an< ^ tr ' a ^ ;t PP eaTS that the fcale of relation is properly 
’ , , , 0 l 4 - 3 *-2-V I -’- Y 
determined; hence 6r- 
1.2 
10 a —15^4-12 c 
I.*-3 
2.3.4 
3 - 4 - 
X x* -f- Sec. —ax 2 —^x-fc.S; and fince 
_ —— . After n 
%—ix +i? jIZT) 2 _ 
terms the feries becomes 2224-1 • 4- 2224-3 . x n +‘ 4“ 
2224-5 • -f- Sec. which arifes from the fraction 
2 72 4“ I . X n -p 2724*3 - X ,! + ‘ -2.2224-1 . X'”'’-' 
ax'~~b.\--\-c— 0, and one value of xmuftbc-, becaufe the 
2 
powers of - are involved in the terms of the propofed fe- 
a b 
Fie4 we have- — • - 4 -c=oo; alfo 6a~-sb-\-ic— 19, and 
2224-1 
to n terms, 
I-2X-J-*' 
' 272 -1 . x" + ‘ 
or 
; hence 1 4 - 3 - v 4 ~. 5 JV ° 4~7 v3 “J” ^ Cc 
.* 4 -* - 272 -j-I . X ”4~222- 
c 7! + > 
