88 ilf?' Hardy, On the convergeyice 



Theorem 4. If ^a,f,u,n ■'*' convergent whenever 'lii,,, is cuii- 

 vergent, then, a,„, is of hounded variation. 



This theorem is due to Hadamard*. We have to shew that, 

 if" X I ^m — f'?n+i j is divergent, v^„ can be so chosen that -n,n is 

 convergent and 1a,nUr,n is not. By Lemma a, we can choose a 

 sequence of positive and steadily decreasing numbers e„, so that 

 e„, — * and Sc,„, where 



is divergent. By Lemma /3, we can then choose the sequence m,- 

 so that %c„j' is divergent. We take 



u^ = U,, tL,n = L\n - ?7,„_, (m > 1), 



where Um,. = 0, 



and U,n = em 



if m^mi, the last expression being interpreted as meaning e„, 

 if a,„ = a^+i • We have then 



mi mi—1 mi-1 



1 1 1 



which tends to infinity with i. Thus l.amUm is not convergent, 

 while ^Um. converges to zero. 



We may call a,,;, a convergence factor if 2o„,m,„ is convergent 

 whenever %Um. is so. Theorems 3 and 4 may then be combined 

 concisely in 



Theorem 5. The necessary and suffi,cient condition that a^n 

 shoidd he a convergence factor is that it should he of hounded 

 variation. 



Double series. 



3. The convergence of a double series, in Pringsheim's sense •]-, 

 does not necessarily involve the convergence of any of its rows or 

 columns |. In this paper I shall confine my attention to con- 

 vergent series whose rows and columns are convergent separately : 

 in this case I shall say that the series is regularly conve7'gent. 

 A regularly convergent double series is also convergent when 

 summed by rows or by columns, and its sum by rows or by colunms 

 is equal to its sum as a double series §. 



Similarly I shall say that a^.^n tends regidarly to a limit if 



lim a,„,, n = «n , lim « „,,, „ = «,„ , 



* I.e. supra. t Bromwich, Infinite Series, p. 72. 



+ Bromwich, ibid., p. 74. § Bromwich, ibid., p. 75. 





