the Summation of certain Classes of Infinite Seriei, 1 5^3 



2 

 equal to the expression marked (l.)> together with —3-— 3> the 



expression just deduced, that is, 



Q — ^ ^ — 2 0/ 3 mp^ 2p + 2 



(a.) 



12(1 +jt>)*... [1 + (m- 1)^ 

 By this general formula any infinite series qf the form 

 1 



S = 



12(1 -^ pf{\ +2pf 

 1 



+ (l +pf{\+2pf{\ +3i;)2... +^^\ 



(^.1 



may be readily summoned, provided only that we previously 

 know the summation of the series S', whose terms differ 

 from those of S by the absence of the final factor in the deno- 

 minator of each. Knowing, therefore, the value of the series 



l^"^ (TT7p+(i +2pf "^ (T+T^^"^ ' • ^''^ 



we may, by successive applications of the formula (at.), pro- 

 ceed, through all the intermediate summations, up to the sum- 

 mation of (6.); and in those cases in which the value of (c.) is 

 unknown, and its approximate summation required,we may ad- 

 vantageously commence by first summing a few terms of the 

 more rapid series (^.), and then, by means of the formula (ff.), 

 gradually descend till we arrive at the proposed form S'= (c). 

 If in the formula (a.) we assume p = 1, we have 



g^j^r2(2^-i)s,_ 3_-i 



7»* \ m 1^ . 2*... wi^j V / 



and if j> = 2, 



When m =zO the sum of the series in each of these cases is 

 known to be 



R2 



