208 

 Proof of theorem 2. 



Suppose that the series ai -\- a^ -\- . . . converges to s, and that 

 the series b\ -\- bi -\- ■ ■ ■ is siimmable of order p (sum =r t) ; Further, 

 let the mean-values of order (/> — 1) of the series b^ -\- b^ -\- . . . be 

 limited. 



We have to prove : 



lim . z=i f! .t 



TIZ^OO 



Put aS^^^ z=: 5„ = s f A,„ then lim . K = and 

 W'O'+i) z= 5(1) Til') + -S(i) TO'^ + . . . + 5^1) 2\i') ... . (6) 



,1 1 H ' 2 ..—1 ' ' n 1 ^ ' 



= (* + /<,) 7'u') + (s + A,) TM ^ + ... + (, + A„) TO') 



= 7i! + S . 



A' = s [Til') + Tip) + . . . + Til')] = sTi]'+i) 



R T(/'+i) 



Aip+^) ' AiP+^) 



n n 



Since the series 6, + 6, + • • ■ is summabie of order p, we have: 

 R 7' (/'+') 



Urn . , , , =: s . lim . -~—.—, =^ s . t. 

 „=«, Aip+^) „=00 Aip+^) 



S = L Tip) + h. Tip) -\ + /,„ TO'). 



* n tl — 1 1 



Since the mean-values of order {p — 1) are limited, it is possible 

 to find M so that: 



Tip) 



~ <iM. Hence: 



Aii>) ^ 



n 



|S|<M[1A,M(;0+ |A,| ^0') + ... + |/i„| yl(")]. . . (8) 

 Put |/«„| =//'!' then we have by (4) 

 \K\Aip)+\K\A^l^^...^\K\Aip) = 



= fl(i) Aip) + Hi'^)Aip) -f . . . + /?(') A'p) 



1 n ' 2 n— 1 ' I) 1 



= ^W^(i) + ^0')41i) + . • . + AO'J AW 



1 n ' 2 n— 1 ' ' H 1 



= Hip) + Hip) + . . . + m')= H(p+i) 



1 ' 2 '1 " 



The inequality (8) may now be written : 



I 5 1 <.1/.i/0'+i) 

 therefore: 



