Professor Hardij. A theorem concerning summable series 307 

 if m <v^m+ Hm. Hence 



m+Hm g m+Hm \ \H 



X {v+l)\v + 2) ^ ^^ »Ti (^ + l)(^ + 2) ~ 4^ 

 when m-^oo . But this plainly contradicts (5). Hence 



lim Sn = 0. 

 It may be shown in just the same way* that 



lim s„ ~ 0. 

 Hence s„-»0, and the theorem is proved. 



* Using a sum S - — — p-^ — --^r • 



