invefii gating the Sums of infinite Series . 
403 
case 2. To find the fitm of the infinite feries - — I — — h 
1 3 • 5 • 7 
\ 2 o 
+ 
4 - See, 
.3 • 5 • 7 • 9 5.7.9.11 
In this cafe « = 1 , in = 2, r = 3, and the 3d differences become 
= o; therefore ^ 4- ^ -f 3c = 2, <7 + 3/; + 15c = 6, ^ -f 5^ 4-35C- 1 2, 
confequently a = 3 = f, c — and hence the fum of the re- 
quired feries becomes — . ? 1 T — . j 1 — _h 
1. 3. 5. 2. 3. 4 3. 5. 2. 2. 2’^. 2. 4 24* 
case 3. "Xi? find the fum of the infinite feries - 
+.5.6.7.$ 
+ 
4. 5.6.7. 8.9 5.0.7.8.9.10 6.7.8.9.10. u 7.8.9.10.11.12 
r Sc.. 
-r 
Here n — 3, m — 1 , r = 5, and the 4th differences become = o ; 
therefore tf-f3^-fi2C-f 60^=1, a -f 4 b 4- 20 c 4- i 20 d — 2, 
a 4- 4- 30 c + 2 i oJ= 4, -f 6^ 42c 4-336^/= 12, confequently 
a =s — 54, £ = 47, c = - 12, d — therefore the fum of the given 
— 46 , 47 12 5 61 
feries is 
.4. 5. 6. 7. 5 4.5.07.4 5.6.7.3 6.7.2.6 50400 * 
case 4. To find the fium of the infinite feries ~ + 
+ 
25 
4-&c. 
4.7.10. 1 3 7.10.13.16 
Here «=i,tfz = 3,r = 3, and the 3d differences are = o ; there- 
fore a+ b + ±c = 1, a + \b+ 28c = 9, + 7^ + 70^= 25, confe- 
fequently , b = ^ ^ = i ; /.the fum of the given feries 
8 4 _ 37 
will be 
1 . 4 . 7 . 3-3*9 4 • 7 • 3 • 2 • 9 7-3*9 22t > d * 
Cg g 2 CASE 
