457 



Hahdy') lias iin estigated several parliculat' exponent-series with 

 particniar metliods that cannot be applied to otiier exponent-series, 

 for instance the series of Fibonacci : 



1,2,3,5,8... 



The onlj general result Hardy could reach was the non-existence 

 of a limit if: 



jOv+i > i . r . pv • %. yt>v . . • . . . • (2) 



but Hardy's example quoted above (where p.j^2''), shows the non- 

 existence of a limit notwithstanding the condition (2) is not satisfied. 

 In the present paper anotlier condition is given (^ 2), with the 

 aid of a theorem of Litti.kwood wliich is treated in § 1. 



^ 1- 



LiTTi.EWOOD has proved the following theorem : ') 



OC 00 



IVieorem 1. If \ 7ia„ j <^ ^, cmd Urn. 2 a,^ x" =z s, then 2 n„ con- 



,(— »-i 1 1 



verges to s. 



For our purpose we want the following extension wliich has also 



been enunciated bj Littlewood : ') 



Tkeorem 2. If the mean-values^) of order k — I of 2! a„ a/'e limi/t'd 



CO 



and liin. 2 a„ x" = s, then .5' a„ is summahle of order k. 



.r—*\ 1 



Littlkwood states that the proof of theorem 2 follows the lines 

 of his proof of theorem 1. The latter being rather long and tedious, 

 it seems not without interest to show that theorem 2 is an immediate 

 consequence of theorem 1. 



Adopting the notation of our article "On a Generalisation of 

 Taubkh's Theorem concerni^ig Poioer Series" '), we have the follow- 

 ing relations between the mean-vakies A,i and the functions (jk- 



1) Quarterly Journal, vol. 38, p. 269, 1907. 



') Proceedings of the London Mathematical Society Ser. 2, Vol. 9, p. 434—448, 

 1911. 



') Proc. of the Lend. Math. See, I.e., p. 448. 



*) For definitions of the mean-values of order p we refer to Bromwich, op. 

 cit., § 122, 123 and Landau, Darstellung und Begriindung einiger neuerei- Er- 

 gebnisse der Funktionentlieorie, § 5. 



'") Proceedings Vol. XXVI (p. 216—225). 



