3*0 
ALGEBRA. 
Ex.-f. Let i-|-- 4---f-- -f- &c. in inf. —S 
2 3 4 
Then - 4 -- + - 4 --+&c. ininf. =S—1 
2 3 4 5 
By ftibtraftion, -A—1—-—|— l —&c. in inf. =1. 
1.2 2.3 3.4 1 
Ex. 2. Ect i -j— — -}- —[- See* in inf. — S 
23 4 
Then - - 4 -- 4 - - 4 -- 4 - See. in inf. — S —- 
3 4 5 6_ 2 
By fnb. ——-j—~—|—-—j—-—^-&c.ininf. —- 
1- 3 2 -4 3-5 4-6 2 
Or ~ ~ + &C. . 
• i -3 2.4 3.5 4-6 4 
Ex. 3. Let ——j—^—j—-—j- &c. 
1.2 2.3 3.4 
Then ——j—-—|—-—&c. —S—- 
2 - 3 3-4 4-5 2 
By Alb. —_!- — 4- 
&c. 
And 
1.2.3 2 - 3-4 3 - 4-5 
1 . 1 
1.2.3 2 - 3-4 3 - 4-5 
1 1 
4- Sec. =2-. 
Ex. 4. To find the fum of the feries 
-f-&c. in inf. 
2.4.6 4.6.1s 
6.8.10 
- \—r 4 -t— + &c. 
2.4 1 4.6 6.8 ' 
Let m—r—i; then the feries becomes- 1 —-—I—iL_ 
1.2 2.3 3.4 
4 - &c. in inf. — 1. After n terms the feries is 1 — 
4 - - j. 4_ &c. whofe fum in infin. is 
« 4 " 2 • ^4"3 ^4-3 • iz 4 " 4 
—-—; therefore, — - — U —-—A ——A &c. to n terms, 2= 
nJ r l x. 2 1 2.3 3.4 
n-\~i n 4-x 
Ex. 6. Let 
Then 
m.m-\-r m-\-r .m-\-2r 
1 1 
+ : 
4 - &C. 
&C. =S~ 
m+r.m+ir 
m.?n-\-r 
By fub. 
vi.m-\-r.vi-\-zr m-\-r ,m-\-2r. ?«4*3r 
Or 
m.m-\-r.m-\-2r m-\-r.m-\-zr. 2224-3r 
4 -&c—- 
2rtn.m-\~r 
l ake away the iaft factor out of each denominator, and 
alTume the refulting feries equal to 5 ; that is, Let 
Let m—r—i ; then- j -(- —--L See. in 
1.2.3 2 - 3-4 3 - 4-5 
inf. =-. After n terms the feries becomes ■ — - - - 
4 n-{-i.n-\-2 . 124-3 
4- — -- j- Sec, in infinitum, whofe fum is 
n-\-2.n-\-2> -224-4 
Then 
By fub. 
And 
4-6 6.8 8.10 
See. =S- - 
2.224-2 • nJ r 2 
: ; therefore, 
2.2.3 2.3.4 
4-&c. to n terms, 
2.4.6 4.6.3 6.8.10 
' r ' • ‘ 
2.4.6 4.6.8 6.8.10 
1 
+ 
m m-^-r 
Then 
Ex. 3. Let — 4- 
m m-\-r m-\-2r 
Sec. =- 
8 
Sec. — —. 
32 
- &c. rzS 
4 2.224- 1 • nJ r 2 
A feries of this kind may he fummed when there are any 
number of factors in the denominator. 
— 4--— 4 -—-4-&c. =5 - 
m-\-r m-\-2r -jr m 
By fub. 
+ &c -=—; 
m.m-^r m-\-r.m-\-2r m-\-2r .m-\-T,r m 
Hence— -1 - 1 4 * ' 1 
a-\-b 
Ex. 7. To find tho fum of the feries - ■ - .— ■ - 
m.m-\-r.n-\-2r 
a-\-2b a 4"3^ 
+ 
■ &c. in infini- 
m-lf-r .m-\-2r ?n-\-2r.m-\-^r. ?«-(-4 r 
See. ——. 
m.m-\-r m-\-r.m-\-2r m J f-2r.m-\-T > r rm turn, by the above examples 5 and 6, 
By Ex. 5 , 
+ 
Or--”? +ar -4- 
m-\-r. m-\-2r 
w 4"3 r 
+ - - - 1 + & c - — 
in-\-2T. 272 - 2-3 r 
_ w + 4r 4- &c. - 
m.m-\-r.m.-\-2r m-\-r.m-\-2r. 7724*3 r ?72 4- 2 7-. 7724 -3 r. 772 4*4'' 
By Ex. 6, "+ r - 4 -- —?±L 
m-\-r 
■ 8 c c. =. 
mr 
1 
mr 
1 
m.m-\-r. 7 >i-\- 2 r m-\-r. 7 n-\-ir. m-\- 2 r, 77 i-\-$r.m-\-<.\.r 
2 mr 
By fub. — — - 
3r - _ - 4 -&c.-- 1 
2 mr 
b 
m.m-\-r.m~\-2r m-{-r.m-\-2r .m-\-m-\-2r.m-\-T,r. 
Mult, by -, - ■ b - - Y . 4 - - ' h . _— 4 - Sec. ■_ „ 
L m.m-\-r ,m-\-2r m-\-r.m^-2r .m-\-2>r m-\-2r.m-\-^r 2ml ~ 
Alfo, 
4 - Sec .: 
2 mr. m-\-r 
m.m-\-r. m-\-2r m-\-r.m-\-2r. m-\- 2 r.m-\- 2 r - mJ r^ r 
By add. - aJrb 4-. - a+2b _- 4- - «+ >* - 4- &c. — 1 J> 
. m-\-2r m-\-r.m-\-2T. m-\-2r.m-j-$r. 2mr r m-\-r 
3 
Let 
