Mathematics. — "(hi Ponder Series of the Form -. xPo — .vPi f .rPs — .." 

 By M. J. Belinfante. (Communicated by Prof. L. E. J. Brouwer.) 



(Communicated at the meeting of April 28, 1923). 



l7itroductio72. 



It is a well-known theorem of Fkobenius that if ^a„ issummable 



..,/'. ■,. 7- •Ï, +*■, + •• • + *■" , , 



of order i, i.e. it hm. =^ s , where .?„ = r/, -\- 



\ n=oo n 



\ 

 -j- rtj -(-... -f- a,, 1 then lini. JS' a,, x" = s, provided .»• approaches 1 



by real values from below (which we denote by ./'— *1). ') 

 Under the same conditions we have:') 



00 



Urn. ^ a„ .r''" — s 

 i_^l 1 



provided /'i <^ /', <C • ■• ^''® integers which satisfy the condition: 



V (p.—p.-i)<i p. . k (1) 



Some condition of the form (1) is necessary as may be seen 

 from the following example, where our condition is broken and 



21 ünX" has no limit as .c— ^1.') 

 1 



Put p., = 2-' and «„^(— l)"+i, then we have: 



,. «,+«,•!-•• s» . 

 Lim. = 4 



n=» '^ 



00 



while -2' a„ ./" = .i- — x'' -\- x* — .r' + • . • oscillates between limits 



1 



at least as wide as 0,498 and 0,502, if .r ^. 1 '). 



Thus we are led to the question : what is the connexion between 

 the exponent-series p„ l>i, }>,, Pt^ fi""! the existence or non- 

 existence of 



lim. (.«#0 — xPi + xVi — . . .) 



') Bromwich, Theory of infinite series, p. 312. 



') Bromwich, op. cit., p. 388. 



'1 Bromwich, op. cit., p. 498 example 30. 



I 



