220 



,.(1), (k)-, Ö1 + Ö2 + .. . + ö„ 



Hu [o ] =^ 



n 



J.k+l) 



71 



_ (4+1) ^n-1 

 On -| — . 



Proof of (17). 



</>^(x) f (1 -..).. ^;^(.r) = 



= .S A-» M„ — ^„_i + (»i + l) . [/4„^1 — ^„ ] — Ji . [/!„ — An^ï]\ 

 ( 1 



= ^ .f" I (« + !). ,4«+i— n . /4« j — [n . An — (w— 1) . A„-i 



I 



a; 1 



1 



■ 'Pk-\ (^)- 



(1 - ..■) . <// (x) = :e .•" 



» 



Proof of (18). 



(«+ 1) . L^!.+i - ^'.f'l - n . [aT - aIHi] I 



= S X" . [<J„4.| — (J,, ]. 

 



^ 3. 

 We prove tlie following extensions of Taubkk's theorem: 



Theorem 2. If l/ni. ii . \_A), — /J„''iiJ = 0, diul |.s'„| <[ c ivliatever be 



11, t/ien 1^ (/,i.i'", ') oscillates as .(■•^■1 'about the same region 

 1 



as An if u — »• GO. 



Theorem 3. \[f //m. m . [ j!,''— ^!,^i] = and liin \È a,,j'' =^ s % 



n=oo 



^//«n ïi'é' liave also : Urn. An = s. 



>.i 1 



Proof of theorem 2. 



.i(') 



From tiie fact that Sn is limited it is "easy^j^to deduce that A„ is 



') See remark '2, at the end of the article. 



