920 



F'or .9, we find by analogous reductions 



E 



y 



From this it may be inferred 



1 1 



Urn IsJ "» <^ — « 



lini |sj 



</^-«, 



since b and jf — ^ may be supposed arbitrarily small. 



As for .s'2, in this sum we cannot assign a majorant-value for 

 the quantity Ljc (y'). But the number of terms of .v, is a fixed one 

 not (lependmq on m. Thus we need only calculate, according to the 

 lemma at the beginning of this paragraph, the required limit for 

 each term separately and then for the whole that limit is in any 

 case not greater. In no one of the lerms the factor Lk{y') depends 

 on ni, so that this factor gives the amount 1 for the required limit 

 and, therefoie, does not intluence it. 



If we further notice that for a given value of k not depending 



i 



on m the limit for //? = c» of mu"^ is also 1, and that, finally, e may 



be chosen arbitrarily small, we infer that 



1 



Urn 1 6-5 1 '" < y — «. 



m =■ 00 



Lastly we consider s^•, substituting k = ni — k', and then omitting 

 again the accent at the letter k, we find 



E-i E-1 .) 



'4 ^ 2i '"k ^ 







The double summation extends over a finite number of terms, 

 which number is independent of m; each of the terms may be 

 identified by fixed values of i and k, likewise independent of m, 

 so that it is sufficient for our purpose to consider the terms separately. 

 To such a term we may apply the inequalities (76) and (78), giving 



mjclik+t^m—k- 



a 



< 



a\[ik-\.[\mk{^—y' i-ey'^-^ 



(y'— «)^+i 



By remarks analogous to those made with regard to the preceding 

 sum we infer from this 



Urn \s. 



r?i= CO 



<^-Y 



