the Summation of certain Classes of Infinite Series. 43 



If the series to be summed were 



1 1 



1*. 3.5.7 + 3*. 5 . 779 + & °'' 



in which p = 2, the process would be this ? viz. 



JL JL A 



2 12 90 



•£ z!_A ^!__L ^!_ JL 7 _ _s 



8 16 4 64- 12 384 1080 ~~ 



8 2 16 3 64 180 



It would be easy, by imitating the steps of the preceding 

 investigation, to deduce a rule for the summation of the infi- 

 nite series 



n(n+p)....(n + mpf + (n + p) ... + (m+ I) j?f + &C * 



This rule would differ from the foregoing in the following 

 particulars, viz. instead of S l9 S 2 , S 3 , &c. we shall have to 

 employ 



Si + — 2 » S 2 + w(w+/ ,j*> S 3 + w(w + i , )(w + 2 ^' &c -> 



and these, instead of being subtracted as before from the se- 

 veral quantities placed under them, must themselves be dimi- 

 nished by those quantities. 



It is clear that when n and p are each unity, as in the first 

 example above, the values which here supply the place of 

 S p S 2 , &c. will be the doubles of these quantities. 



As an example, let the series 



1 1 



1 . 2 . 3 . 4 2 + 2 . 3 . * . 5* + " &C * 



be proposed, in which w = 1, p ;= 1, m = 3, 



■ JL _L 



2 9 



6 2 ""6 12 " T08^36 =S " m ' 



6 6 2 36 12 



