86 Mr Hardy, On the convergence 



On the convergence of certain multiple series. By G. H. Hardy, 

 M.A., Trinity College. 



[Received 15 May 1917.] 



1. In a paper published in 1903 in the Proceedings of the 

 London Mathematical Society*, and bearing the same title as this 

 one, I proved a theorem concerning the convergence of multiple 

 series, of the type 



which is given (with an improvement in the conditions) on p. 89 

 of Dr Bromwich's Theory of infinite series. This theorem is one 

 of a class of some importance ; and I propose now to state and 

 prove the leading theorems of this class in a form more systematic 

 and general than has been given to them before. I shall begin by 

 recapitulating, with certain changes of form, some known theorems 

 concerning simply infinite series ; and I shall then obtain the 

 corresponding theorems for double series in a form as closely 

 analogous as possible. The generalisation from double series to 

 multiple series of any order may well be left to the reader. • 



Simply infinite series, 



2. I shall say that a function a.,„,, real or complex, of a positive 

 integral variable m is of hounded variation if 



■^ i (^m ~ f'-m+l I 

 1 



is convergent. It is plain that this condition involves the existence 

 of a = lim a^n- 



Theorem 1. The necessai^y and sufficient condition that a^n 

 shoidd be of bounded variation is that its real and imaghuiry ptarts 

 should be of bounded variation. 



This follows at once from the inequalities 



I ^m ^m+l I ^ j Cini (^m+l \ ; j Pm Pm+l i ^ ! ^^ in ^^9n+l |> 

 I (^m C^7n+i I ^ I ^m ^m+i \ "i' [ Pm Pm+i \ > 



where a,,,, = «,„ + i^,„. 



* Ser. 2, vol. 1, pp. 124 — 128. See also 'Note in addition to a former paper on 

 conditionally convergent multiple series', ibid., vol, 2, 1904, pp. 190 — 191. 



