306 Professor Hardy. A theorem concerning summable series 



(A) // Xctn is summable (C, 1), and In^ | ajp+' is convergent 

 for some positive p, then Sa„ is convergent 



(B) Ifta^ is summahle {G, 1), and either (a) a„ is real and 



K 



an>--, 



then Sa„ {5 convergent. 



6. To prove (A) we observe that 



^«+i "" , ,7X1 = -^ (^ + 1) ^+1 .(z^+l)2'+ia 

 and so* 



= {\n~^~T)^i\ (1) = [-] . 

 Hence (w + 1) 6^+i-> 0, and so, by (3), s^-^ A. 



_ 7. To prove (B) we observe first that, if it is condition (B) that 

 IS given, we may suppose without loss of generality that a, is rm^ 

 for we may treat the real and the imaginary parts of the series 11 

 separately. But then (^) becomes a special case of (a). It is there- ^^ 

 tore only necessary to consider condition (a). Further, we may 

 plamly suppose that ^ = 0. ^ 



Suppose that lim s^ = \ > 0, 



and choose a sequence of values of n for which 



Let us denote a value of n, belonging to this sequence, by m: 

 and choose H so that i}<KH < i\. Then ' ^ ' 



4 

 Su = s,y^ + a^+i + a^+2 + . . . 4- di 



Vm + 1 v) ^ m+l 



>h'K-KH>\\, 



By the well known inequality 





