78 Mr Kienast, Proof of the equivalence of 



and therefore the conditions of Stolz's theoreDi are satisfied. Thus 

 we can state: 



Lemma 7. //' lim h^"^ = H exists and is finite, then 



lim 2 d,^X'^/ I d^,. = H. 



V. Let ai = l, ao = a3=---=0 (6); 



then , sl'^=l (n = l, 2, ...), 



^^=^^^ (n=2,3, ...); 



therefore lim s^^^^ = 1, and consequently 



lims;f=lims^^'>=... = l. 



Furthermore h^^^ = h^^^ = h'-^^ ^ . . . = 1 (n = l, 2, ...)• 



From (1) and (6) we obtain 



C ) \n-K 



Imi s\'^'= 1 = hm Cn,K - hm - S dx.K, 



n-^cc n-*-oo 7j->-Qo n 1 



1 n—K 



or lim - 1 dK,K = 0. 



VI. Now passing in equation (1) to the limit ?i-^qo , we find 



, , , , /I n-K \ (n-K , . ,n-K ) 



limsf = lim/i2«-lim (- S < J lim 2 d.,,/ii''7 S d,,. \, 



and this equation leads to the theorems: 



Theorem 3. Whenever lim h^^\^ exists and is finite, then 



lim s^"' exists too and has the same value. 



n 



More generally 



Theorem 4. When the function lf^^_^ oscillates between finite 

 limits, then s^^ oscillates between the same limits. 



VII. The reverse propositions can be established in the same 

 way. From the definitions follow 



n 



h^''=- 



J^^{(XH-2)4!>,-(X + l).f>J 



+ ^{(.+ 2).f>,-(. + 2)0} 



(n+^y J2) 1 I ^ + 2 (2) 

 n(n + l) «+2 ^nKZiXiX + 1) ^+2" 



