42 Prof. Young's Investigation of Formula? for 



1 l 



n(n+p)...(n + mp) (n+p) [n + (m+l)p'] C ** 



then we have the following rule for the summation of the infi- 

 nite series : 



\ L ! L & C 



«*(w+p) (n + mp) T (n + p)*(n + 2p)...[n + (m+l)py ' 



From I take Sj and divide the result by p ; from the quo- 

 tient take S 2 and divide the result by 2p; from the quotient 

 take S 3 and divide the result by Sp; and so on, till the di- 

 visor becomes m p 9 which will furnish a quotient equal to 

 the sum of the proposed series. 



As to the values of the subtractive quantities, S lf S 2 , S 3 , 

 &c, they are at once obtained from the fundamental relation 

 with which we set out. Thus, 



s, = -L 



1 p n 



1 s, 



s* = 



2p n (n + p) 2 (n + p) 



g = 1 = SSg 



3pn(n+ p) (n + 2p) 3 (n + 2p) 



... m [n -J- (m — l)pY 



As an example let it be required to find the sum of the in- 

 finite series 



+ ~* s , 5 + ..,,,.. + &c. 



I 2 . 2 . 3 . 4, 2 2 . 3 . 4 . 5 T 3 2 . 4 . 5 . 6 



in which w = 1, p = 1 and w = 3. 



The values of S l9 S 2 , S 3 , are, in this case, 



S» m 1, S 2 = T , S 3 = -, 



hence, arranging these in a row and prefixing the subtractive 

 sign, the operation by the rule will be as follows : 



"~T """is 



