404 
Mr. vince’s new Method of 
case 5. 'To find the Sum of the infinite Series T p 
1 • 3 - S ’ 7*9 ' 
+ 
10 
+ 
20 
3.5. 7.9. n 5.7.9. 11. 13 7.9. 11. 13 15 
In this cafe n = 1 , m - 2, r = 4, and the 4th differences become 
= 0; therefore a + b + ic+ 1 $d= 1, a + 3^ + 1 5c + 105^ = 4, 
a + 5 & + 35 c + 3 l 5 d= 10, ^ + ; £ + 63(7 + 693'’ = 20, confe- 
quently £ b - _ 3 _, d — .^, and hence the fum of 
the given feries becomes ^ 1- 3 1 
1.3.5.7.2.4.16 3-5.7.2.3.16 
1 1 £ — =_£_ 
5.7.2.2.16 7 . 2 . 48 364 * 
This proportion may alfo be applied to find the fum of all 
thofe feries whofe numerators being unity, the denominators 
fhall be deficient by any number of correfponding terms, how- 
ever taken : for as the product of all fuch factors muff form a 
progreflion, whofe differences will become equal to nothing, 
if fuch products be affumed for the numerators of the given 
feries having its factors compleated, another feries will be 
formed equal to the given feries, whofe fum can be found 
by this propofition. 
> 
case i. To find the fum of the infinite feries i— 3- 
I » 2 • A. « O 
2 • 3 • 5 • 7 3. 4.0.6 
; + &C. 
T l 
* *< = 
By completing the fadlors in the denominators, and multi- 
. . , , . ; .*. • . ’ . 
plymg the numerators by the fame quantities the given feries 
r. + 
oecomes 
it 
— t _i_ . — 
1 • 2 . 3 • 4 • 5 * ^ 2v. 3 « ’4 • S *-6 * 7 3 4 5 4 6 . 7” + 
in which cafe n = 1 , = 1 , r = 5, and the 3d differences become 
w* f. .ft % ^ Tf u * 1 
^ 2 * 
+ 
35 
