204 



Farther we may ask for a condition, sncli that the prodnct of a 

 convergent series by a series whicli is siimmabie of order p 

 and satisfies the condition, will also be snnunable of order p. This 

 can be seen from : 



Theorem 2 : The product of a convergent series by n series 

 which is sunuiKthle of order p and whose riienn-vahles of order p — 1 

 are limited, is summable of order p. 



Finally we consider (he product of two series, wliicii are summable 

 respectively of order p and q ; then we are led to : 



Theorem 3 : The product of a series lohich is summable of order 

 p and lohose mean-values of order p — 1 are limited, by a series 

 luhich is summable of order q, is summable of order p -\- q. 



If we call a series which is summable of order p (p>i) and 

 whose mean-values of order p — I are limited, joinable of order p, 

 and if we call an absolutely converging series joinable of order 

 zero, then the above theorems are included in the following: 



Theorem -. The product of a series which is joiiiabh of order p, 

 bij a series irhich is summable of order q, is summable of order 



The proofs however will be given separately. For the sake of 

 completeness we begin with the deduction of some well-known 

 formulas. 



Let x(i), .r*'i) .r(^), .... be an arbitrary sequence of complex 



numbers; we define: 



.,,•(2) = .»(i) + «(1) 4- -f a-(i) ....... (1) 



.r(i-Hi) = «(fc) 4- xW + + xW .... (2) 



We denote .r<^' by ^W if .j;(.i'=:l, whatever be /. It is easy to 

 I •' i I 



verify that : 



(« l-A— 2)/ 

 AM=-^ — '— (3) 



(71— l).'(;fc— 1).' 



We consider the series «i -|- «2 + •• ■ ■ a'ld b[ -\- b^ -\- ■ ■ ■ ■ with 



their product: Ci-}-C2 + (where c„=:aib„-\- -|- ö„ *i), and 



we put : 



5(1) = s„ = a, -f a, + . . . + fl„ . j 



.... (3a) 



2'U) = t„~b, + !>, + ... + b„ . ' 



WO^ = M'„ = C, + C, + . . . + f„ . 



