304 Professor Hardy. A theorem concernirvg summahle series 



A theorem concerning summahle series. By Prof. G. H, Hardy. 

 [Received 23 December 19:^0. Read 7 February 1921.] 



1. It is well known that if the series Sf/„ is summahle (C, 1) 

 that is to say if s^^An (1). ' 



•where s„ = «„ + «! + ••• + a,^, s^ =So + s,+ ... + s^, 



then S '^^ =-° + ^h + (.J. 



n + 1 1^2^ ^■^' 



is convergent*. The converse is not true, as may be seen at once 

 from trivial instances to the contrary. It is therefore interesting 

 to frame a theorem of this kind which embodies a necessary and 

 sufficient condition for summability. Such a theorem is the 

 following. 



Theorem. The necessary and sufficient condition thatta„ should 

 he summahle {C, 1) to sum A is that 



Sr,+{n + l)hn+i-^A (3), 



where h„ = — ^^ + -£zi±L + (a\ 



2. It is plain that we may (replacing a^ by a,,- A) suppose 

 without loss of generality that ^ = 0. If 



s'^ = {n), s„ = s\ - s'„_i = (n). 

 Again, (3) and (4) involve the convergence of (2), i.e. involve t^-^B, 

 where a„ = (?i + l)c„ and i^ = Co + C:+ ... + c„. And 



n n-1 



Sn = 2 (v + 1) C, = {n +l)t.„- t t, = (71). 

 



Hence we may suppose 5„ = o (n) in proving either part of the 

 theorem. 



This being so, we have 



7 _ v^ "^^ "^"-i _ 1- „ V ^^ ^"-1 



n+1 V+ i- TO^x n+1 V + 1- 



= lim('~^^'^ '- +T -^-'^ ] 



\m + l n+2 n+i(v+l){v-\-2)J 



m-H«.co 



S ^ Q 



n + 2 ^^+l(^/^-l)(^. + 2)' 



s, + {n+l)h^^, = -'l +(n + l) S . T^^ ^^• 



n + 2 \^-^(jj+i)(j, + 2) 



* H. Bohr 'BicTrag til cle Dirichlet'ske Rackkers Theori,' Inaugural Dissertation 

 (Copenhagen, 1910), p. 100. The theorem follows at once from (1) and the identity 



n~2 



v + l {'' + l){v + 2)(p + 3)^ n(n + l)^ n + 1 



