invefiigating the Sums of infinite Series 405 
= 0; therefore a+b+ ic — 1;, a + zb+ 6 i = 24, a 4- 3^4- 120 = 35, 
coniequentiy = 8, b — 5, c = 1 , and therefore the fum of the ie- 
8 V 
ries required is ■ — 
+ 
4* 
21 1 
2 • 3 • 4 • 5 • 5 2 • 3 • 4 • 5 • 4 3 - 4 - 5 *3 7 ^°° 
case 2. To find the fum of the infinite fries - — 4- 
+ 
3 • 7 • *3 5 • 9- *5 
' 4 - &c. 
This feries, when completed, becomes — \- 
r 1.3.5.7.9.11* 
495 , >ooi , 2755 + &c< 
4- 
3 • 5 • 7 • 9 • 1 1 • *3 5 - 7 • 9 • 1 1 * 1 3 - J 5 7 •. 9 • 1 1 • J 3 • l 5 • 1 7 
where »=i, w= 2, r=5, and the 4th differences are=o; 
therefore # 4 - /> 4- 3c 4- 1 = 1 89, a 4- 3 b 4- 1 5c 4- 105^ = 495, 
a + 5^ + 3 S C 4-315^=100 1, <24-7^ + 6304-693^= 1755? con- 
sequently ^ = 96, ^ = 48, 0 = 10, o?= 1 ; and hence the fum of 
96 
+ 
48 
+ 
10 
the given feries is — 
& 1 » 3 * 5 - 7 - 9* 2 -5 3. 5 • 7 « 9 • 2 . 4 5 • 7 • 9 • 2 • 3 
1. 487 
+ 
7 . 9 . 2 . 2 1^900 
- ' j 
case 3. To yW the fum of the infinite fries - — — ^ + 
4- 
4 . 7 . ib . 19 7 . 10 . 19 . 22 
4* & c. 
This feries pefolves itfelf into 
70 
1 . 4 . 7 . 10 . 13 . 16 
+ 
r jO ’ 
+ 
208 
7 — 4- &e. where n = 1 ? 
4 . 7 . iO . 13 . 16 . 19 7 , lO y 13 . 16 . 19 . 22 
w = 3, r=5, and the 3d differences' =0 ; therefore a+b-t- 4 ^= /°> 
a + 4^ + 2*80 = 130, <z + 7^ + 700 = 208, from whence tf = 54 > 
bz=zt2, therefore the fum of the given feries is 
cE * 5 ^- 
