206 



tlieorem tlie snmmability of order /) (and .also the convergence) of 

 a series implies the siimniabilitj' of order {p -\- i) if j'^l'). 



Proof of theorem 1. 



Suppose that the series ai -|- «2 + ■ • • • 'S absolutely convergent 

 (suiii = A'); let the series 6i -|- 62 + • ■ • ■ be summable of order />, or 



Urn. ^ t, then we have to prove: 



lim . = s . t 



„=00 Aii'+i) 



n 



Now we liave : 



Vr(p+i) = ai 7'f/'+i) + a2 nP+^) + .. . + a„ TC/'+i) . . (7) 



n u n — 1 1 



Put /(„ ^ t, then lim h,, = 0. Substitution of 



^(p+i) 



n 



n n n 



in (7 gives : 



n 



n n n — 1 " — I 1 1 



=: t [ai ^(/^+1) + a2 Aip+^) + . . . -f a„ ^(/'+')] + 



'- n n — 1 1 



+ [«1 ^O'+i) K + 12 ^(/H^i) A„_i + ...+«„ A[p+^)hi] . 



= P+ Q. 



F= t [ai A<^P+^) + 02 AiP+h -^ . . . + a„ ^^^Z'+DJ = 



■- n n — 1 1 



= t [«1 Aip+^> + («2 - »i) ^l/Hj') + • • • + («« - s,.-i) ^0'+»)J = 



= t\si (AiP+i) —A<.f+^i) + s2 (A(.P+^)^A(i'+/))+ ... + s„ A[p-^^)] = 



■- ^ n n — 1 ' " — 1 11^2 1 



= t l«i A(P) + S2 Aip) ^ ^ . . + s„ A[P)^ 



= t rS(i) A(p) + SO) Aip) + ... + SW A(p)] 



= t \Sip) ^c) + «(/-^ ^(1)' + . . . + Sipt Am 



•-1 H ' 2 n— 1 ' ' 11 1 -■ 



= t [SW + S(p) + . . . + S(p)] 

 = t S(p+i). 



n 



P .SCp+i) 



Hence lim -— rrr = t . lim . -^~r— = i .s for öi + «2 + . . . converges 



absolutely and is consequently summable of order p. 

 ») Bkomwich. I.e. (p. 312). 



