MONTMORT. 129 



Montmort takes credit to himself for summing this series, so as 

 to find its value when a and h are large numbers ; but, without 

 saying so, he assumes that a = 4. Thus the series becomes 



4l&f|c-l \g- ^ |c — *7 



c 



\h_ ' |5-3 ' \h-Q 



Let p = h + ^, then c=p+l] thus the series within brackets 

 becomes 



+ (p-(i)(p->7)(p-8) + ... 



Suppose we require the sum of n terms of the series. The 

 r^^ term is 



(p-Sr + S) (^-'3r + 2) (^^-3r + l) ; 



assume that it is equal to 



where A, B, C, D are to be independent of r. 

 We shall find that 



A=j>{p- 1) {p - 2), 

 B=-(9/-4.5;>+()0), 

 (7=54p-216, 

 i>=-162. 



Hence the required sum of n terms is 



np (p - 1) {p - 2) - '^^^ (V- ^op 4- 60) 



n{n-l){n-^) __ n {n - 1) {n - 2 ) (7^ - S) 



^ 1.2.3 K'^W--^^) 1.2.3.4 ^'" 



This result is sufficiently near Montmort's to shew that he must 

 have adopted nearly the same method ; he has fallen into some 

 mistake, for he gives a different expression for the terms inde- 

 pendent oip. 



In the problem on chances to which this is subser\dent we 



should have to put for ?i the greatest integer in -^ . 



9 



