223 



that [ / 1 <^ f if ts<C'i'<CJ- t'lirllier '\t is possible to calculate 

 -^s ^ ?j so '''f^t l-'^ — *'| <C ^ if §«<C''-'<C^> for we have: 



J. «(!-«)' 



.r 1 



loff + 



-loq 



§. 



= s + 



{l-x)log 





•(1 i») lofi 



l-a, ' • 1-5, 1-§J 



and the expression between brackets tends to zero as .c-^i. 



In like manner we can calculate s^S, so that \lll\ <C ^f if 

 ï<^.);<^l. Combining these results we have if § <^ .;; <^ 1 : 



1 / |< 6, I // — .V |< 6 and I /// t < 26, 

 therefore : 



Since e is arbitrary and //m. C {I — .(-■) =r we. infer: 



lini. (f (x) = s. 

 We now prove theorem III as follows: b^y h^ypothesis we have 



X) 



lim. q>, (d:) ^ /i,in. 2(7,1 x" ^ s; applji'ig the lemma we get: 



Urn. (f^ (j;) :=: a ; Ihn . (f\ {x) = ■« ; Urn. (f (x) zrz 3 ; 



or: 



lim.2:[AV — j(,''-l\x" =, 



Moreover we have bj hy|)othesis: 



Ihn. n [A^p'^—AT-i] = 



and therefore by Tauber's original theorem'): 



liin. An ^ s 



¥vom Urn. n[A';!'^ — ^ili] = 0, (13) and (14) we deduce: 

 Hence by (22) 



(22) 



Ihn. YAi'~'^ - A'I\} = 



Um. An = f, 



Remark 1. 



It is not difficult to see that the following statement is an im- 

 mediate consequence of theorem 3 : 



') Lim. nan = and Um.. San x» = s imply Um. l,ai =s. 



1 =; 00 X ^ 11 H = _ 1 



