EXPANSION PROBLEMS. 



391 



from a certain point on. When n is large, therefore, 



Jf (x) Vn (x) dx — \ col 

 L \ ^Pn 



— 1 \ 9+2 / _ X \ 3+2 



g-a.p„,r_|_^2 



OO^Pn 



< 



3c 



Pn 



9+3' 



iPnTi 



Asq ^ 2 (mod. 3), the expression in brackets reduces to 



9+2 / \ /_ IN 9+2 ^2 



N( 



f J f e-'-Pn'^+e-'^P"'^) = 2( J e^''"'^cos — pnTT. 



V3 (\ V3 \ (\ V3 



cos — pnTT = C0s[-+ W+ — enJTT = ± COS f - + — - fn ] TT, 



lim 



71= 00 



V3 



cos — PnTT 



V3 

 2 ■ 



It follows that as soon as n is sufficiently large, the absolute value of 

 the bracket is greater than 



3 i 



a nd hence that from a certain point on 



j^f(.x)Vn(,x) 



dx 



> 



1 



Pn 



9 + 2' 



2« 



ipnir 



With regard to the magnitude of the denominator in (13), it suffices 

 to observe that, when pn is real and positive, 



\un{x)\^3 e^""'', I Vn (x) \ ^ 3 e^'""('^- ^\ 



and so 



I Jo Un (x) Vn (x) dx \ ^ 9 IT 6^""^. 



This inequality, with the one obtained for the numerator, shows 

 that as n becomes infinite 



\an\ > 



Ci 



p.«+2' 



where Ci is positive and does not depend on n. 



From this relation it is apparent that the general coefficient does not 

 approach zero rapidly enough to enable the terms of the series l^anUn{x) 



