Mathematics. — "A Generalisation of Mkktens' Theorem'". Bj 

 iM. J. Bei.infantk. (Comiminicated by Prof. L. E. J. Brouwer). 



(Communicated at the meeting of February 24, 1923). 



The theory of infinite series, whieii ho far chiefly consisted of 

 convergent series, being extended to the so-called summable and 

 asymptotic series, it is natural to generalize as much as possible 

 the classical results about convergent series to these classes of series. 



For the well-known theorem of Mertkns this has been done by 

 Hardy (Bromwich, Theory of Infinite Series, p. 284), who used 

 BoREi/s method of summation. In the present paper we treat a 

 somewhat different generalisation, whereby we are only concerned 

 with Cesaró's method of summation. 



The product or the product-series of the series 



a, -h a, -f- and ^, -f />, 4- 



is defined as the series <•,+«, + .... 

 where c, = a, bi -\- a, 6,_i +....-}- a, 6,. 



CesarÓ has proved: if two series are convergent, their product 

 will be summable of order 1, and if two series are summable 

 respectively of order p and q, their product will be summable of 

 order [p + q -\- 1) '). 



If we call a convergent series summable of order zero, then the 

 first part of Cesaró's theorem is included in the second. 



Mehtens' theorem, which runs as follows. "If one of two convergent 

 series converges absolutely, their product is convergent" may now 

 be stated thus : 



The product of a series which is absolutely convergent, by a 

 series whicii is summable of order p is summable of order zero. 



In the first place we are led to the following generalisation: 



Theorem 1 : The product of a series which is absolutely convergent, 

 by a series which is summable of order p, is summable of order p. 



') Bromwich. Theory of infinite series, § 125 pp. 314—316. 



14 



Proceedings Royal Aoad. Amsterdam Vol. XXVI. 



