MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 285 



Also r (! + ,) = 0-8856... =770-', say. 

 So that if 



and if, further, 



fi, r(i + O<r(i + ). 



Returning to the definition of b n , we get from these results 



1 6. |* A, A, 2 



r = 



[(i +17.) r (/*+ 1) + 77, 2 r (r + 1) r {(->) *+!}], 



r 1 



where 



77, = T; O if X'u < 1 and A', ?* L, 

 and 



77, = 1 if either or hoth of the conditions i 1 , A-, = Ic 3 are satisfied. 



Now consider F(f) = r(^+l)r{(nf )!+!}, 7a function of a continuous real 

 variable If /> denote the logarithmic derivate of the gamma function, we have 



AF(f) = A- 



d * I 



But, since -^^(f) = 2 TZ - n , we see that iA (f ) increases with ^ when > 0. 

 T ' 1 



Hence -^F() is negative when : ?i f ; and therefore, in the summation 



r= I 



the terms decrease until the middle term (or terms) and then increase, terms 

 equidistant from the beginning and end of the summation being equal. 



Two distinct cases now come under our consideration, (I.) when ^ 1, (II.) when 

 <!. 



In the first case, using the equality 



we have, from the result just proved, 



2 r(r +l)r{(n-r) +l} ^(n-1) r(A- +l) T {(7i-l)& + 1}, 



< (71 - 1 ) T (nk + I ) \ (I- a 1 *) 



_' A 



r= 1 



