1^8 Dr Kienast, Extensions of 



Thus we obtain 



Theorem 4. If Km s\';^ = I exists and is finite, th 



en 



Mni^a^x" = I. 



X-»-l 1 



The first condition of Theorem 3 is therefore necessary. 



6. To demonstrate the rest of the assertion in Theorem 8, it 

 follows from the hypothesis lim \/^^'^ = that Theorem 11 is 

 applicable. Thus the assumptions are transformed into 



lim 2 - ^rl^+'\-=.i 

 lim-7^;^+^'=0. 



71-*- 00 ^i 



From this last equation follows 



I '' 1 <>.^^^ .. 1 



.(A+l) 



lim:^S -^r(^+^'=lim^^^ = 



,,-^00 nx+lK + l « ,,^^ 71 + 1 



Hence Theorem 2 can be applied to the series 2 - x'' • 



11 , • • n{n + \) ' 



and the conclusion is that 



lim t -r~~^, = I. 

 Theorems 12 and 13 now yield 



lim 4^^=/, 



with which the proof of Theorem 3 is completed. 



7. The foregoing deductions are valid for X = l, 2,.... For 

 X = they still hold, except those in § 6. This case requires the 

 proof of the following special case of theorem 2 : 



Theorem 14. // 



limS^-i--r^V = /, 

 ^^1 1 n(n + 1) « 



lim ~r^'^ = 



I 



