225 



A somewhat different generalization of theorem B has iieen given 

 b^- A. KiKNA.sT '). KiKNAST defines : 



" (1) 



2 ah r„ := ^ i ai- 



1 



n 1 



'^+1) 1 "-J^ n) 0+1, ""1 1 o.) 



n ,,4-1 ; A- 



and proves the following theorem : 



.1 f'+'^ . «> 



Theorem C: The conditionx lim. - r„ := and ////^ JS'a,, .t" = 5 



are each necessanj for the e.i'/'sieiice of I/'m. s„ =.v, and taken toge- 



nrxoo 



iher they are sufficient. 



Tlie mean- val nes".?,, differ from Cksaro's or Höldeh's mean-valnes, 

 bnt in a second paper") Kiknast has shown the eqnivalenoe of liis 

 mean-valnes with those of Cesaró-Höldeh. 



Remark 2. 



We have tacitly assnnied that ^fa,,.!'" converges if — 1 <^ .r <^ 1. 



This is however superflnons for our purpose as the condition 



Uni.n\^An — An-i '" ] = implies the convergence of ^ ^/„a-" pro- 

 pje» I 

 vided |.»r<J. 



Indeed from Ihn.n [^« — ^n— 1] = we infer tiie absolnle 



convergence of '/-^(a-) =r ^ [/l,f — A','-x\x" provided .rj <^ 1. 



I 



Further we have by (17): 



therefore the absolute convergence of ifiX'V), which implies the abso- 

 lute convergence of <i'jc[x), im|ilies also the absolute convergence of 

 Vi—i (.I-)- Repealing the aigument we infer the absolute con\ergence 



as 



ot r/, (.r) := ^ a„«" provided \x\<^\. 

 1 



1) Proceedings of tiie Cambridge Phil. Soc, vol. 19 (1918), p. 129. 

 "') Proceedings of the Cambridge Pliil. Soc, vol. 9,0 (1920), p. 74. 



