142 Br Kienufit, Eidenaioiiii of 



shall exist, or at any rate that the functions under the limit sign 

 shall oscillate finitely, I procffcd to prove 



Theorem 16. TJie two cunditions 



CO 



lim 2 ««*■" = I (Jinite), 



x^l 1 



M-S-OG hi 2 fA— 1 



are each necessary for the existence of the limit 



lim s^'^ = l: 



n 



mid, taken together, they are sufficient. 



It is not possible to demonstrate this theorem for every set of 

 numbers 6^. The following example shows this. 



Mr Riesz has pointed out* that 



1 'i, 1 



hm ^ - Sk 



WH-Qo log n 1 K 



exists and is finite in the case of 



However, Abel's limit 



lim 2 Kr''^~'^^x'^ 



x^\ 1 



does not exist, as the function behaves like 



r(«)(iogi)"' 



when *■— »1. 



].l. The demonstration depends on some identities analogous 

 to those employed in the case of the arithmetic means, viz. 



* See G. H. Hardy, ' Slowly oscillating series', Proc. London Math. Soc, ser. 2, 

 vol. 8 (1910), p. 310. 



