VOL. LXXII.] PHILOSOPHICAL TRANSACTIONS. 313 



and leave a series whose numerators are unity, and whose denominators want the 

 middle factor. 



Part ii. Prop. — To find the sum of the infinite series 

 I + —1 — + I + &c 



n (» 4- m) n + rm n + m n 4- (; • + 1 ) m ' n + 2m . . . . n + (r + 2) m ' 



when the last differences of the numerators become equal to nothing. 



Assume a + nb -\- n (n + m) . c + n (n + m) . (n -f 2m) . d + &c. to any 

 number (?•') of terms; then, if for n we write n + m, n -f- 2m, n + 3m, &c. 

 successively, there will result a series of quantities whose r'th difference is = 0; 

 substitute therefore this series of quantities for p, q, s, &c. respectively, and the 

 given series becomes of a form which manifestly resolves itself into several other 

 series, the number of which is r ; and the sum of each of these being taken by 

 a well known rule, the sum of the given series becomes 

 f I h - I i 



n(n + vi)....(n + (r—i))}i).m.r n + m....(n+ (r—\)7ii).m.(r— \) ' n + 2m....(n+ (r— i)m).m(r— 2) 

 ■j- &c. where the law of continuation is manifest. 

 Exam. To find the sum of the infinite series 



— - -A- -— - -I- 10 i _21— _i_ & 

 1.2.3.4 - *" 2.3.4.5 *" 3.4.5.6*" 4.5.6.7 ' ' 



Here n = 1, m = 1, r = 3, and the 3d differences become = 0; therefore 

 a + b + 1c = 3, a + lb + 6c = 6, a + 3b + 12c = 10, consequently a = l, 

 6=1, c = -i., and therefore the sum sought will be 



FTs .3.3 ~^~ 2.3.2 ~*~ 2T3 — 36* 



This proposition may also be applied to find the sum of all those series whose 

 numerators being unity, the denominators shall be deficient by any number of 

 corresponding terms, however taken: for as the product of all such factors must 

 form a progression, whose differences will become equal to nothing, if such pro- 

 ducts be assumed for the numerators of the given series having its factors com- 

 pleted, another series will be formed equal to the given series, whose sum can 

 be found by this proposition. 



Exam. To find the sum of the infinite series 



1.2.4.6 ' 2.3.5.7 ' 3.4.6.8 



By completing the factors in the denominators, and multiplying the nume- 

 rators by the same quantities, the given series becomes 



1.2.3 15 4.5.6 + 2.3.4 24 5.6. l + 3 .4 . /.'ft . 7 ■ 8 + &C ' in W ' lich CESe " = »• 

 m = 1, r = 5, and the 3d differences become = 0; therefore a -\- b + 2c = 15, 

 a + lb + 6c = 24, a + 3b + 12c = 35, consequently a = 8, b = 5, c = 1, 

 and therefore the sum of the series required is 



5 l _ 211 



o Q 1 <; a I Q a. si ~~ 70 



1.2. o. 4. 5. 5 r 2.3.4.5.4 ' 3.4.5.3 7200 



By this proposition we may also investigate the sum of the series when there 

 vol xv. S s 



