dijferent mean values 79 



Continuing this process we find 



A;:^=^n,«^-;i+' ^aai in 



In the same way as we determined Ch,k by (2) and (3), we 

 obtain here 



n + K-\-l 



Thus, since 



6)1 K+] — 6j; 



n + 1 



_ (n + kY 



^"' " " n{n + l)...{n + K-l) ' 



Taking the values (4) for the numbers a^, we find from (7) 



f = X f (^ + l)(^+2)...(X + /c) { X+fcY ] 



■■''~ \ V X(\+l)...(\ + «-l)J ■ 



The considerations in § IV show that 



Expansion of /a_x in descending powers of X gives 

 A. = g(«-1)«(« + 1)- + ^, + ...; 



n 



thus lim S /a, K = 00 . 



W-*.cc \ = 1 



Introducing in equation (7) the vahies (6) for the numbers a^, we 

 find 



1 = lim e„^« + lim - 2 /a,« lim 2/a,«sJ;''L/ S/a^^ ; 



which gives, on account of Stolz's theorem, 



lim- i/A,. = 0. 



Thus equation (7) is completely determined and leads to 



Theorem 5. Whenever lim s^^'L exists and is finite, then 

 lim h^"^ exists too and has the same value. 



n 



Theorem 6. Whenever the function s^^^^ oscillates finitely, 

 then h^n oscillates between the same limits. 

 Theorems 3 and 5 together constitute theorem 2. 



