459 



^ 2. 



We now prove tiie following theorem 



. '•»+( . , ...... k\—l 



Theorem 4. If I < ^, < ^^ < /t„ and /.-, > J + ^^^^7-^, then 



r„ 2k, 



f{x) == «'■» — «;'i-|- .•(;'■■■' — .... doe.i n t tend to <i Uiitit ((.s a;—*l '). 



Proof: We show that the .series of coefficients of f{.v) (wliich 



consists of the terms 1, (?•; — r, — 1) zero^, — 1, (r, — r^ — 1) zeros, 1, 



and so on...) is not siimmable of the fiist oi'der, i.e. that (/„does 



not tend to a limit as h— >-30 . Then it is impossible that /'(.(■) should 



tend to a limit as .v—^1, for this implies by theorem 3 the existence 



of lim. ö„ '). 



We show that rj„ does not tend to a iiniii if /i —*• oo, by calculating- 

 two positive numbers y and m so that: 



We have : 



_ ^1 + ^, + • • • + g« na^ -\- (w— 1) a, + • • + [n — {n-l)] a„ 



_ r-ip— [r2^, — >■! + 1] + [>'2// —i-o + 1] — • • • + L'"2p — np + IJ 



V'ip 



ri — r<i-\-r3—... — r-ip r-^p — r-ip-i -f r2p-2 — . . . — r , 



Tip 



f2p 



■v+1 



Since 1 <^ /(-•,< —^ < /(-•,, it follows that r.,^i > r., Ic^ and 



r,-|-i — rv>(^i — 1) r,. Substituting this in the e.xpression for <j,„ 

 we have : 



(^■l-l)r.2/._l + (^-1— 1)»>-3 4- • . . + (ii-l)n 



or < 



'2/) 



< 1 



^1 — 1 r2p-[ + '•2p— 3 4- • • • + ''I 



< 1 — 



^2 

 ^1 — 1 



ko 



^2p-l 

 ■11 1 - 



1 H 1 h • • • H 



< 1 



ki — 1 

 ko 



k^P 

 2 



1 — 



/fc2 

 2 



') We suppose r,, = 1. 



') The condition s„i <c is satisfied since s„ is i or 0. 



Proceedings Royal Acad. Amsterdam. Vol. XXVI- 



30 



