286 MB. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



From this we deduce* that 



(2) 



In the second case, let m be the greatest integer such that *, < 1. 

 When n * 2 + 1, we have, as in the first case, 





("since the integrand is less"! 

 S (?< - 1 ) T (nkt + 1 ), than or e q ua l to uni ty ] 



so 



that, when* n S 2m+l, we deduce from the inequality on the preceding page : 



lastly, when n 2: 2m +2, we have 



2 1 



r- I 



= 22 r(rJt +i)r{(i-r)^+i}+ n T"r(^ +i)r{(w-r)A ;o +i} 



r=l 



^2 2 



since r(r*.+ l)<l when lsrs=m, and the largest terms of the summation 

 2 are the first and last. 

 But, when 1 == r S m, r{()i-r) k a + 1 } S T (*+ 1) ; and hence 





r(rt,+l)r{(n-r)*,+l} 



< 2mr(n*b+l) + (n-l- 



[since l- 



and consequently, in the second case, for all values of n, we have 



We see at once that this formula covers the first case also (m being then zero) 

 * This is only proved when nil; but it is obviously true when n = 0. 



