209 



II <M— 



Since Hm . H'^^ ^ O and hence /zm . — ^,-- := O, it follows iliat 



^^ 



«=0= ^(''+1^ 



Hence : 



^^</'+^) . R S 



Urn . =. lim . -— — h hm . ; — - = *■ . t. 



„=„ A(f+i) „=300 A(f+i) „=„ AU'+^) 



n n n 



Proof of theorem 3. 



Suppose that the series a\ -\- n« -\- . . . is siimmable of order /), 

 and that its mean-values of order (/) — 1) are limited, tiien we have : 



S(v+^) 1 Sip) I 



lim . " ^ ^ 8 and — 7^<r a fixed number M. 



„=„ ^(;'+i) Ai^') ^ 



Let h\ -\- b'i -\- . . . be summable of order q, or 



lim . — --,-— = t. 



We have to prove ; lim . —, — ; — r^ = 4 . t 



n 1 II ' 2 n— 1 ' ' n 1 ^ ' 



Put — r-rr> = ^ + '''». then lim . h„ = 



W(p+i+^) = 51/') [tA(9+i) j-h ^(?+i)] + . . . 4- 51/') [« ^(?+i) 4- h ^(v+i)] 

 = «[«(z-) ^(9+1)+...+ 5W^(?+i)l + [*§(/>) A 4(v+i) + ...+ .S(/')A ^(?+i)] 



'1 n ' 'n 1 ' ' ^ 1 ti n ' 'nil-' 



C^= < [S^p) ^(7+1) + 5(/')^)?+i) 4- . . . + Sip) A(9+m 



■-1 '1 ' 2 II— 1 ' '11 1 -' 



= t [S(p+J) + S^p+i) + . . . + S(P+<))] = t . .?(/'+'?+i) 



TJ S(/'+ï+l) 





40^+9+1) Aiv+q+\) 



