of certain multiple series 87 



Theorem 2. The necessary and sujjicient cunditiuu that a real 

 function a,„ shoidd be of bounded variation is that it should he of 

 the form A^n—AJ, where A.^. and A J are 'positive and decrease 

 steadily as m increases. 



The sufficiency of the condition follows at once from the 

 •inequality 



I Clm ~ ftjn+l ! ^ \A,n, — -4,„ + i) + {Ajn — A ,„,+]). 



In order to prove that it is necessary, let us suppose that a,„ is 

 of bounded variation, and let us write 



P„i = ! flm - (I'm+l I (an, " ««<+! > 0), p,„ = (a,,,, - a,„+, < 0), 



Pm'^ ! "m - (Im+l \ (('in " «/»+! < 0), j)„/ = (a,;, - d ,„^.^ > 0), 



B = ^' i) 7? ' = S « ' 



-'J??; — —' /'n> -"m -^ J'n • 



II! VI 



Then B,,, and i?„/ are positive and decrease steadily as m in- 

 creases ; and 



B,n - BJ = 2 ((f„ - a„,+i) = ff,,, - a. 



in 



We may therefore take A,„,= B„,, + G and ^j,/= 5,,,' + C, where 

 C and C" are suitably chosen constants. 



Theorem 3. // a,,; is of bounded variation, and Su,,,, is con- 

 vergent, then 2rt,„M„,, is convei^gent. 



Theorem 1 shews that it is enough to prove this theorem 

 when a„, is real. Theorem 2 shews that it is enough to prove it 

 when «„, is positive and steadily decreasing. In this form the 

 theorem is classical*. 



Lemma a. If 2c,„ is a divergent series of positive terms, we 

 can find a, sequence of positive numbers e,„, tending steadily to the 

 limit zero, such that 2e,„c,„ is divei'gent. 



Lemma /3. If 2c,„, is a divergent sei'ies of positive terms, we 

 can find a sequence of integers m^ such that the series Sc,„' , where 

 Cm' = if m = nil and c,,' = c,,, otherwise, is divergent. 



Lemma a is due to Abelf. Lemma (3 is quite trivial, and the 

 proof may be left to the reader. 



* See Bromwieb, Infinite Series, p. 48. Theorem 3 is given by Dedekind in bia 

 editions of Diiicblet's Vorlesungcn iiber Zahlcntheorie : see e.g. p. 255 of the tbird 

 edition. Tbe central idea of all sucb tbeorems is of course Abel's. Tbe line of 

 argument followed bere is due substantially to Hadamard, 'Deux theoremes d'Abel 

 sur la convergence des series'. Acta Matheinatica, vol. 27, 1903, pp. 177 — 184. 



t 'Sur les series', (Euvves, vol, 2, pp. iy7-~20o. 



