211 

 Hence we have if n is sufficiently great: 



n 



Since e ^ is arbitrary we have: 



V 



lim . — — -; — r— = 0. 



Hence : 



Urn . =^ lim -I- lim : — ■ — :=r s . t 



Remark 1. 



K. Knopp ') ad S. Chapman') iiave limited the order of snminability 

 of the product of two series, whicii are snmmable of order /* and 

 q, by considering non-integral orders of summability. It may happen 

 that the theorems proved above give more result, as is seen by the 

 following example; 



The series 1 — 1 -(- 1 — 1 + . . . is summable of order 1 and its 

 mean-values of order are limited. Hence, applying theorem 3, we 

 see that the product of this series by a series which is summable 

 of order p, is summable of order [p -\- 1). Now the so-called index 

 of summability of the series 1 — 1 + . . . is zero (see Chapman, I.e.); 

 the index of a series which is summable of order p, cannot exceed 

 p: hence the index of the product cannot exceed p-\-\, and there- 

 fore we can only infer by Chapman's theory that the product is 

 summable of order p -\- 2. 



Remark 2. 



Hahdy ') has also given the following extension of Mehtens' theorem 

 which is totally different from the generalisations mentioned above, 

 and which contains Mektens' theorem as a special case : 



If Sfl,, is absolutely convergent and Sè„ is a finitely oscillating 

 series whose »"' term tends to zero, then their product is a finitely 

 oscillating series, and if the limits of oscillation of 2 b,, are ^^ and 

 (J, those of the product are s. (s, and s. ^,. 



') Silzuagsberichte der Berliner Math. Gesellschaft 19U7 (p. 1—12) 



') Proceedings of the London Mathematical Society, Ser. 2 Vol. 9 (p. 369— 409). 



') Proceedings of the London Mathematical Society. Ser. 2 Vol. 6 (p. 410— 423). 



