t 41 ] 



VIII. Investigation of Formula for the Summation of certain 

 Classes of Infinite Series. By J. R. Young, Professor of 

 Mathematics in Belfast College. 



[Continued from vol. x. p. 124, and concluded.] 



JET us take A of the form 



n (n + p) (n + 2p) ... (n + nip) 



then, by the general relation referred to, we may substitute 

 for it the expression 



j_r 1 



mp L n (n + p) (n + 2p) ... [n + (m— l)p] 



-J \. 



H + 2 p) ... in + mp)J 



n* l (n+p) (// + 2p) ... (n + mp). 



Now, by the same relation, we may for the first expression 

 within the brackets substitute 



1 r 1 



/ 



(m -\)p ^ n * ^ n + p \ ( W + 2p\... [ n +(m- 2)p] 



_^ I 1 



n l (n + p){n + 2p).» [n + (m - l)p]f 



and for the first of these we may make a like substitution ; so 

 that by proceeding in this manner we shall at length arrive at 



_L r l 1 i 



2 P \n*(n+p) n*~ l (n + p) (n + 2p) J 

 and finally at 



L/JL 1 1 



P W n K ~ 1 {n+p)J 

 whence a rule may be easily deduced for the summation of a 

 series of the proposed form, provided certain subordinate se- 

 ries can be summed. As an illustration of this suppose x = 2, 

 and put for abridgement 



1? + (n+p)* + (n + 2py + &c * = *. , 



n{n + p) 7 (n+p) (n + 2p) T {n + 2p) (n + 3p) TOtc — ?l 

 1 1 



n(n+p)(n + 2p) + (n+p) (» + 2p)(n + Sp) + &C '- S * 



Third Series. Vol. il. No. 64. July 1837. G 



